Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions...

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Variants of T X Cross-connections Subsets and partitions Reg(T θ X ) Biordered set Summary Cross-connections and variants of T X Azeef Muhammed P. A. 1 1 Institute of Natural Sciences and Mathematics, Ural Federal University, Ekaterinburg, Russia. 2 2. My sincere thanks to A. R. Rajan, University of Kerala, India, and M. V. Volkov, Ural Federal University, Russia, for their support and guidance. Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of T X 1 / 11

Transcript of Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions...

Page 1: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Cross-connections and variants of TX

Azeef Muhammed P. A.1

1Institute of Natural Sciences and Mathematics,Ural Federal University, Ekaterinburg, Russia.

2

2. My sincere thanks to A. R. Rajan, University of Kerala, India, and M. V. Volkov,Ural Federal University, Russia, for their support and guidance.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 1 / 11

Page 2: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

The talk will be on the variants of the full transformationsemigroup in the context of Nambooripad’s cross-connections.

The full transformation semigroup TX is the semigroup of allmappings on a non-empty set X.A variant T θ

X of the full transformation semigroup (TX , ·) for anarbitrary θ ∈ TX is the semigroup T θ

X = (TX , ∗) with the binaryoperation

α ∗ β = α · θ · β where α, β ∈ TX .

In 2015, Dolinka and East explored the structure of T θX , its

idempotent generated subsemigroup, its regular part, its idealsetc.The following subsets of TX was crucial in their discussion,

P1 = {a ∈ TX : aθ R θ} P2 = {a ∈ TX : θaL θ}.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 2 / 11

Page 3: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

The talk will be on the variants of the full transformationsemigroup in the context of Nambooripad’s cross-connections.The full transformation semigroup TX is the semigroup of allmappings on a non-empty set X.

A variant T θX of the full transformation semigroup (TX , ·) for an

arbitrary θ ∈ TX is the semigroup T θX = (TX , ∗) with the binary

operationα ∗ β = α · θ · β where α, β ∈ TX .

In 2015, Dolinka and East explored the structure of T θX , its

idempotent generated subsemigroup, its regular part, its idealsetc.The following subsets of TX was crucial in their discussion,

P1 = {a ∈ TX : aθ R θ} P2 = {a ∈ TX : θaL θ}.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 2 / 11

Page 4: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

The talk will be on the variants of the full transformationsemigroup in the context of Nambooripad’s cross-connections.The full transformation semigroup TX is the semigroup of allmappings on a non-empty set X.A variant T θ

X of the full transformation semigroup (TX , ·) for anarbitrary θ ∈ TX is the semigroup T θ

X = (TX , ∗) with the binaryoperation

α ∗ β = α · θ · β where α, β ∈ TX .

In 2015, Dolinka and East explored the structure of T θX , its

idempotent generated subsemigroup, its regular part, its idealsetc.The following subsets of TX was crucial in their discussion,

P1 = {a ∈ TX : aθ R θ} P2 = {a ∈ TX : θaL θ}.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 2 / 11

Page 5: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

The talk will be on the variants of the full transformationsemigroup in the context of Nambooripad’s cross-connections.The full transformation semigroup TX is the semigroup of allmappings on a non-empty set X.A variant T θ

X of the full transformation semigroup (TX , ·) for anarbitrary θ ∈ TX is the semigroup T θ

X = (TX , ∗) with the binaryoperation

α ∗ β = α · θ · β where α, β ∈ TX .

In 2015, Dolinka and East explored the structure of T θX , its

idempotent generated subsemigroup, its regular part, its idealsetc.

The following subsets of TX was crucial in their discussion,

P1 = {a ∈ TX : aθ R θ} P2 = {a ∈ TX : θaL θ}.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 2 / 11

Page 6: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

The talk will be on the variants of the full transformationsemigroup in the context of Nambooripad’s cross-connections.The full transformation semigroup TX is the semigroup of allmappings on a non-empty set X.A variant T θ

X of the full transformation semigroup (TX , ·) for anarbitrary θ ∈ TX is the semigroup T θ

X = (TX , ∗) with the binaryoperation

α ∗ β = α · θ · β where α, β ∈ TX .

In 2015, Dolinka and East explored the structure of T θX , its

idempotent generated subsemigroup, its regular part, its idealsetc.The following subsets of TX was crucial in their discussion,

P1 = {a ∈ TX : aθ R θ} P2 = {a ∈ TX : θaL θ}.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 2 / 11

Page 7: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

They gave the following diagram to show how a typical D-class ofTX (in the left), breaks up to the corresponding D-classes of T θ

X .

In this talk, we discuss the ideal structure of Reg(T θX)—the

regular part of T θX (i.e., P1 ∩ P2).

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 3 / 11

Page 8: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

They gave the following diagram to show how a typical D-class ofTX (in the left), breaks up to the corresponding D-classes of T θ

X .

In this talk, we discuss the ideal structure of Reg(T θX)—the

regular part of T θX (i.e., P1 ∩ P2).

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 3 / 11

Page 9: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Even if a semigroup is regular, its variant need not be regular.

But, Khan and Lawson (2001) had shown that if a semigroup isregular, the regular part of its variant forms a semigroup.Hence, Reg(T θ

X) is a regular subsemigroup of T θX .

So, in the discussion regarding the structure of Reg(T θX), it is

natural to look at known structure theories of regular semigroups.In the area of structure theory of regular semigroups, there aretwo constructions using categories : both due to Nambooripad.The first one uses the idempotent structure of the semigroup(AMS Memoirs No. 224, 1979) and it belongs to the realm of thecelebrated Ehresmann-Schein-Nambooripad theorem.The second approach using the ideal structure of the semigroupwas initiated by Hall (1973) and Grillet (1974).In 1994, Nambooripad (1994) extended the latter approach toarbitrary regular semigroups using cross-connected categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 4 / 11

Page 10: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Even if a semigroup is regular, its variant need not be regular.But, Khan and Lawson (2001) had shown that if a semigroup isregular, the regular part of its variant forms a semigroup.

Hence, Reg(T θX) is a regular subsemigroup of T θ

X .So, in the discussion regarding the structure of Reg(T θ

X), it isnatural to look at known structure theories of regular semigroups.In the area of structure theory of regular semigroups, there aretwo constructions using categories : both due to Nambooripad.The first one uses the idempotent structure of the semigroup(AMS Memoirs No. 224, 1979) and it belongs to the realm of thecelebrated Ehresmann-Schein-Nambooripad theorem.The second approach using the ideal structure of the semigroupwas initiated by Hall (1973) and Grillet (1974).In 1994, Nambooripad (1994) extended the latter approach toarbitrary regular semigroups using cross-connected categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 4 / 11

Page 11: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Even if a semigroup is regular, its variant need not be regular.But, Khan and Lawson (2001) had shown that if a semigroup isregular, the regular part of its variant forms a semigroup.Hence, Reg(T θ

X) is a regular subsemigroup of T θX .

So, in the discussion regarding the structure of Reg(T θX), it is

natural to look at known structure theories of regular semigroups.In the area of structure theory of regular semigroups, there aretwo constructions using categories : both due to Nambooripad.The first one uses the idempotent structure of the semigroup(AMS Memoirs No. 224, 1979) and it belongs to the realm of thecelebrated Ehresmann-Schein-Nambooripad theorem.The second approach using the ideal structure of the semigroupwas initiated by Hall (1973) and Grillet (1974).In 1994, Nambooripad (1994) extended the latter approach toarbitrary regular semigroups using cross-connected categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 4 / 11

Page 12: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Even if a semigroup is regular, its variant need not be regular.But, Khan and Lawson (2001) had shown that if a semigroup isregular, the regular part of its variant forms a semigroup.Hence, Reg(T θ

X) is a regular subsemigroup of T θX .

So, in the discussion regarding the structure of Reg(T θX), it is

natural to look at known structure theories of regular semigroups.

In the area of structure theory of regular semigroups, there aretwo constructions using categories : both due to Nambooripad.The first one uses the idempotent structure of the semigroup(AMS Memoirs No. 224, 1979) and it belongs to the realm of thecelebrated Ehresmann-Schein-Nambooripad theorem.The second approach using the ideal structure of the semigroupwas initiated by Hall (1973) and Grillet (1974).In 1994, Nambooripad (1994) extended the latter approach toarbitrary regular semigroups using cross-connected categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 4 / 11

Page 13: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Even if a semigroup is regular, its variant need not be regular.But, Khan and Lawson (2001) had shown that if a semigroup isregular, the regular part of its variant forms a semigroup.Hence, Reg(T θ

X) is a regular subsemigroup of T θX .

So, in the discussion regarding the structure of Reg(T θX), it is

natural to look at known structure theories of regular semigroups.In the area of structure theory of regular semigroups, there aretwo constructions using categories : both due to Nambooripad.

The first one uses the idempotent structure of the semigroup(AMS Memoirs No. 224, 1979) and it belongs to the realm of thecelebrated Ehresmann-Schein-Nambooripad theorem.The second approach using the ideal structure of the semigroupwas initiated by Hall (1973) and Grillet (1974).In 1994, Nambooripad (1994) extended the latter approach toarbitrary regular semigroups using cross-connected categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 4 / 11

Page 14: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Even if a semigroup is regular, its variant need not be regular.But, Khan and Lawson (2001) had shown that if a semigroup isregular, the regular part of its variant forms a semigroup.Hence, Reg(T θ

X) is a regular subsemigroup of T θX .

So, in the discussion regarding the structure of Reg(T θX), it is

natural to look at known structure theories of regular semigroups.In the area of structure theory of regular semigroups, there aretwo constructions using categories : both due to Nambooripad.The first one uses the idempotent structure of the semigroup(AMS Memoirs No. 224, 1979) and it belongs to the realm of thecelebrated Ehresmann-Schein-Nambooripad theorem.

The second approach using the ideal structure of the semigroupwas initiated by Hall (1973) and Grillet (1974).In 1994, Nambooripad (1994) extended the latter approach toarbitrary regular semigroups using cross-connected categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 4 / 11

Page 15: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Even if a semigroup is regular, its variant need not be regular.But, Khan and Lawson (2001) had shown that if a semigroup isregular, the regular part of its variant forms a semigroup.Hence, Reg(T θ

X) is a regular subsemigroup of T θX .

So, in the discussion regarding the structure of Reg(T θX), it is

natural to look at known structure theories of regular semigroups.In the area of structure theory of regular semigroups, there aretwo constructions using categories : both due to Nambooripad.The first one uses the idempotent structure of the semigroup(AMS Memoirs No. 224, 1979) and it belongs to the realm of thecelebrated Ehresmann-Schein-Nambooripad theorem.The second approach using the ideal structure of the semigroupwas initiated by Hall (1973) and Grillet (1974).

In 1994, Nambooripad (1994) extended the latter approach toarbitrary regular semigroups using cross-connected categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 4 / 11

Page 16: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Even if a semigroup is regular, its variant need not be regular.But, Khan and Lawson (2001) had shown that if a semigroup isregular, the regular part of its variant forms a semigroup.Hence, Reg(T θ

X) is a regular subsemigroup of T θX .

So, in the discussion regarding the structure of Reg(T θX), it is

natural to look at known structure theories of regular semigroups.In the area of structure theory of regular semigroups, there aretwo constructions using categories : both due to Nambooripad.The first one uses the idempotent structure of the semigroup(AMS Memoirs No. 224, 1979) and it belongs to the realm of thecelebrated Ehresmann-Schein-Nambooripad theorem.The second approach using the ideal structure of the semigroupwas initiated by Hall (1973) and Grillet (1974).In 1994, Nambooripad (1994) extended the latter approach toarbitrary regular semigroups using cross-connected categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 4 / 11

Page 17: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

For this, he introduced rather sophisticated notions like normalcones, factorisations, normal duals, local isomorphisms,cross-connection bifunctors, transpose of morphisms etc.

In this situation, classifications and descriptions ofcross-connections in special classes are of sufficient interest.Various classes like transformation semigroups, lineartransformation semigroups, their singular parts, inversesemigroups, completely 0-simple semigroups, etc have beenstudied earlier by Rajendran(2000), Rajan and Azeef(2016).In this talk, we discuss the cross-connection structure ofReg(T θ

X)—the regular subsemigroup of the variant T θX .

The purpose is two fold.First, this semigroup provides a concrete setting where all theabstract notions of cross-connection theory has transparent, yetnon-trivial meanings.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 5 / 11

Page 18: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

For this, he introduced rather sophisticated notions like normalcones, factorisations, normal duals, local isomorphisms,cross-connection bifunctors, transpose of morphisms etc.In this situation, classifications and descriptions ofcross-connections in special classes are of sufficient interest.

Various classes like transformation semigroups, lineartransformation semigroups, their singular parts, inversesemigroups, completely 0-simple semigroups, etc have beenstudied earlier by Rajendran(2000), Rajan and Azeef(2016).In this talk, we discuss the cross-connection structure ofReg(T θ

X)—the regular subsemigroup of the variant T θX .

The purpose is two fold.First, this semigroup provides a concrete setting where all theabstract notions of cross-connection theory has transparent, yetnon-trivial meanings.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 5 / 11

Page 19: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

For this, he introduced rather sophisticated notions like normalcones, factorisations, normal duals, local isomorphisms,cross-connection bifunctors, transpose of morphisms etc.In this situation, classifications and descriptions ofcross-connections in special classes are of sufficient interest.Various classes like transformation semigroups, lineartransformation semigroups, their singular parts, inversesemigroups, completely 0-simple semigroups, etc have beenstudied earlier by Rajendran(2000), Rajan and Azeef(2016).

In this talk, we discuss the cross-connection structure ofReg(T θ

X)—the regular subsemigroup of the variant T θX .

The purpose is two fold.First, this semigroup provides a concrete setting where all theabstract notions of cross-connection theory has transparent, yetnon-trivial meanings.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 5 / 11

Page 20: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

For this, he introduced rather sophisticated notions like normalcones, factorisations, normal duals, local isomorphisms,cross-connection bifunctors, transpose of morphisms etc.In this situation, classifications and descriptions ofcross-connections in special classes are of sufficient interest.Various classes like transformation semigroups, lineartransformation semigroups, their singular parts, inversesemigroups, completely 0-simple semigroups, etc have beenstudied earlier by Rajendran(2000), Rajan and Azeef(2016).In this talk, we discuss the cross-connection structure ofReg(T θ

X)—the regular subsemigroup of the variant T θX .

The purpose is two fold.First, this semigroup provides a concrete setting where all theabstract notions of cross-connection theory has transparent, yetnon-trivial meanings.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 5 / 11

Page 21: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

For this, he introduced rather sophisticated notions like normalcones, factorisations, normal duals, local isomorphisms,cross-connection bifunctors, transpose of morphisms etc.In this situation, classifications and descriptions ofcross-connections in special classes are of sufficient interest.Various classes like transformation semigroups, lineartransformation semigroups, their singular parts, inversesemigroups, completely 0-simple semigroups, etc have beenstudied earlier by Rajendran(2000), Rajan and Azeef(2016).In this talk, we discuss the cross-connection structure ofReg(T θ

X)—the regular subsemigroup of the variant T θX .

The purpose is two fold.

First, this semigroup provides a concrete setting where all theabstract notions of cross-connection theory has transparent, yetnon-trivial meanings.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 5 / 11

Page 22: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

For this, he introduced rather sophisticated notions like normalcones, factorisations, normal duals, local isomorphisms,cross-connection bifunctors, transpose of morphisms etc.In this situation, classifications and descriptions ofcross-connections in special classes are of sufficient interest.Various classes like transformation semigroups, lineartransformation semigroups, their singular parts, inversesemigroups, completely 0-simple semigroups, etc have beenstudied earlier by Rajendran(2000), Rajan and Azeef(2016).In this talk, we discuss the cross-connection structure ofReg(T θ

X)—the regular subsemigroup of the variant T θX .

The purpose is two fold.First, this semigroup provides a concrete setting where all theabstract notions of cross-connection theory has transparent, yetnon-trivial meanings.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 5 / 11

Page 23: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Second, we give an alternate path to the structural description ofReg(T θ

X) given by Dolinka and East, using subsets and partitionsof X.

This, in turn suggests, that their results obtained in the specificcase of this variant semigroup, is much more universal in nature.This discussion also yields a descriptions of the biorder structureof Reg(T θ

X).It can be seen that the principal ideals (or equivalently Green’srelations) in TX and its variants are determined by the subsetsand partitions of X.So, naturally, the description of the ideal structure of thesesemigroups involves subsets and partitions.For that, we borrow the terminology of Dolinka and East.Let A be a subset of X and α an equivalence relation (or apartition) on X.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 6 / 11

Page 24: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Second, we give an alternate path to the structural description ofReg(T θ

X) given by Dolinka and East, using subsets and partitionsof X.This, in turn suggests, that their results obtained in the specificcase of this variant semigroup, is much more universal in nature.

This discussion also yields a descriptions of the biorder structureof Reg(T θ

X).It can be seen that the principal ideals (or equivalently Green’srelations) in TX and its variants are determined by the subsetsand partitions of X.So, naturally, the description of the ideal structure of thesesemigroups involves subsets and partitions.For that, we borrow the terminology of Dolinka and East.Let A be a subset of X and α an equivalence relation (or apartition) on X.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 6 / 11

Page 25: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Second, we give an alternate path to the structural description ofReg(T θ

X) given by Dolinka and East, using subsets and partitionsof X.This, in turn suggests, that their results obtained in the specificcase of this variant semigroup, is much more universal in nature.This discussion also yields a descriptions of the biorder structureof Reg(T θ

X).

It can be seen that the principal ideals (or equivalently Green’srelations) in TX and its variants are determined by the subsetsand partitions of X.So, naturally, the description of the ideal structure of thesesemigroups involves subsets and partitions.For that, we borrow the terminology of Dolinka and East.Let A be a subset of X and α an equivalence relation (or apartition) on X.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 6 / 11

Page 26: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Second, we give an alternate path to the structural description ofReg(T θ

X) given by Dolinka and East, using subsets and partitionsof X.This, in turn suggests, that their results obtained in the specificcase of this variant semigroup, is much more universal in nature.This discussion also yields a descriptions of the biorder structureof Reg(T θ

X).It can be seen that the principal ideals (or equivalently Green’srelations) in TX and its variants are determined by the subsetsand partitions of X.

So, naturally, the description of the ideal structure of thesesemigroups involves subsets and partitions.For that, we borrow the terminology of Dolinka and East.Let A be a subset of X and α an equivalence relation (or apartition) on X.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 6 / 11

Page 27: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Second, we give an alternate path to the structural description ofReg(T θ

X) given by Dolinka and East, using subsets and partitionsof X.This, in turn suggests, that their results obtained in the specificcase of this variant semigroup, is much more universal in nature.This discussion also yields a descriptions of the biorder structureof Reg(T θ

X).It can be seen that the principal ideals (or equivalently Green’srelations) in TX and its variants are determined by the subsetsand partitions of X.So, naturally, the description of the ideal structure of thesesemigroups involves subsets and partitions.

For that, we borrow the terminology of Dolinka and East.Let A be a subset of X and α an equivalence relation (or apartition) on X.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 6 / 11

Page 28: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Second, we give an alternate path to the structural description ofReg(T θ

X) given by Dolinka and East, using subsets and partitionsof X.This, in turn suggests, that their results obtained in the specificcase of this variant semigroup, is much more universal in nature.This discussion also yields a descriptions of the biorder structureof Reg(T θ

X).It can be seen that the principal ideals (or equivalently Green’srelations) in TX and its variants are determined by the subsetsand partitions of X.So, naturally, the description of the ideal structure of thesesemigroups involves subsets and partitions.For that, we borrow the terminology of Dolinka and East.

Let A be a subset of X and α an equivalence relation (or apartition) on X.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 6 / 11

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Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Second, we give an alternate path to the structural description ofReg(T θ

X) given by Dolinka and East, using subsets and partitionsof X.This, in turn suggests, that their results obtained in the specificcase of this variant semigroup, is much more universal in nature.This discussion also yields a descriptions of the biorder structureof Reg(T θ

X).It can be seen that the principal ideals (or equivalently Green’srelations) in TX and its variants are determined by the subsetsand partitions of X.So, naturally, the description of the ideal structure of thesesemigroups involves subsets and partitions.For that, we borrow the terminology of Dolinka and East.Let A be a subset of X and α an equivalence relation (or apartition) on X.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 6 / 11

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Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

• We say the subset A saturates thepartition α if each α-class contains atleast one element of A.

• The partition α separates A if eachα-class contains at most one elementof A.

• So, the subset A is a cross-sectionof the partition α, if A saturates αand α separates A.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 7 / 11

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Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

• We say the subset A saturates thepartition α if each α-class contains atleast one element of A.

• The partition α separates A if eachα-class contains at most one elementof A.

• So, the subset A is a cross-sectionof the partition α, if A saturates αand α separates A.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 7 / 11

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Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

• We say the subset A saturates thepartition α if each α-class contains atleast one element of A.

• The partition α separates A if eachα-class contains at most one elementof A.

• So, the subset A is a cross-sectionof the partition α, if A saturates αand α separates A.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 7 / 11

Page 33: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Now, this leads us to define a category Pθ with the set of objectsas :

vPθ = {A ⊆ X : πθ separates A}.

A morphism between two subsets A and B in Pθ is any functionf from A to B.This category may be shown to be a normal category.A normal category essentially characterises the principal idealsof a regular semigroup.In other words, any regular semigroup naturally determines twonormal categories.So, in the case of Reg(T θ

X), the second category is determinedby the partitions of X saturated by Im θ, say Πθ.The cross-connection construction also involves certainintermediary regular semigroups arising from these categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 8 / 11

Page 34: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Now, this leads us to define a category Pθ with the set of objectsas :

vPθ = {A ⊆ X : πθ separates A}.

A morphism between two subsets A and B in Pθ is any functionf from A to B.

This category may be shown to be a normal category.A normal category essentially characterises the principal idealsof a regular semigroup.In other words, any regular semigroup naturally determines twonormal categories.So, in the case of Reg(T θ

X), the second category is determinedby the partitions of X saturated by Im θ, say Πθ.The cross-connection construction also involves certainintermediary regular semigroups arising from these categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 8 / 11

Page 35: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Now, this leads us to define a category Pθ with the set of objectsas :

vPθ = {A ⊆ X : πθ separates A}.

A morphism between two subsets A and B in Pθ is any functionf from A to B.This category may be shown to be a normal category.

A normal category essentially characterises the principal idealsof a regular semigroup.In other words, any regular semigroup naturally determines twonormal categories.So, in the case of Reg(T θ

X), the second category is determinedby the partitions of X saturated by Im θ, say Πθ.The cross-connection construction also involves certainintermediary regular semigroups arising from these categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 8 / 11

Page 36: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Now, this leads us to define a category Pθ with the set of objectsas :

vPθ = {A ⊆ X : πθ separates A}.

A morphism between two subsets A and B in Pθ is any functionf from A to B.This category may be shown to be a normal category.A normal category essentially characterises the principal idealsof a regular semigroup.

In other words, any regular semigroup naturally determines twonormal categories.So, in the case of Reg(T θ

X), the second category is determinedby the partitions of X saturated by Im θ, say Πθ.The cross-connection construction also involves certainintermediary regular semigroups arising from these categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 8 / 11

Page 37: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Now, this leads us to define a category Pθ with the set of objectsas :

vPθ = {A ⊆ X : πθ separates A}.

A morphism between two subsets A and B in Pθ is any functionf from A to B.This category may be shown to be a normal category.A normal category essentially characterises the principal idealsof a regular semigroup.In other words, any regular semigroup naturally determines twonormal categories.

So, in the case of Reg(T θX), the second category is determined

by the partitions of X saturated by Im θ, say Πθ.The cross-connection construction also involves certainintermediary regular semigroups arising from these categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 8 / 11

Page 38: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Now, this leads us to define a category Pθ with the set of objectsas :

vPθ = {A ⊆ X : πθ separates A}.

A morphism between two subsets A and B in Pθ is any functionf from A to B.This category may be shown to be a normal category.A normal category essentially characterises the principal idealsof a regular semigroup.In other words, any regular semigroup naturally determines twonormal categories.So, in the case of Reg(T θ

X), the second category is determinedby the partitions of X saturated by Im θ, say Πθ.

The cross-connection construction also involves certainintermediary regular semigroups arising from these categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 8 / 11

Page 39: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Now, this leads us to define a category Pθ with the set of objectsas :

vPθ = {A ⊆ X : πθ separates A}.

A morphism between two subsets A and B in Pθ is any functionf from A to B.This category may be shown to be a normal category.A normal category essentially characterises the principal idealsof a regular semigroup.In other words, any regular semigroup naturally determines twonormal categories.So, in the case of Reg(T θ

X), the second category is determinedby the partitions of X saturated by Im θ, say Πθ.The cross-connection construction also involves certainintermediary regular semigroups arising from these categories.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 8 / 11

Page 40: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

In Reg(T θX), they are isomorphic to the sets P1 and P2 discussed

earlier, seen as subsemigroups of TX .

Via cross-connections, Nambooripad also gave the explicitrelationship between the principal ideals of a regular semigroup.For that, he used the notion of a normal dual of a normalcategory and a cross-connection functor.In Reg(T θ

X), this functor, say Γθ, is completely characterised bythe sandwich element, θ.Thus, we can realise Reg(T θ

X) as a cross-connection semigroup

(Πθ,Pθ; Γθ) = {(θa, aθ) : a ∈ Reg(T θX)}.

This representation gives the following description of thebiordered set and sandwich sets of Reg(T θ

X).

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 9 / 11

Page 41: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

In Reg(T θX), they are isomorphic to the sets P1 and P2 discussed

earlier, seen as subsemigroups of TX .Via cross-connections, Nambooripad also gave the explicitrelationship between the principal ideals of a regular semigroup.

For that, he used the notion of a normal dual of a normalcategory and a cross-connection functor.In Reg(T θ

X), this functor, say Γθ, is completely characterised bythe sandwich element, θ.Thus, we can realise Reg(T θ

X) as a cross-connection semigroup

(Πθ,Pθ; Γθ) = {(θa, aθ) : a ∈ Reg(T θX)}.

This representation gives the following description of thebiordered set and sandwich sets of Reg(T θ

X).

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 9 / 11

Page 42: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

In Reg(T θX), they are isomorphic to the sets P1 and P2 discussed

earlier, seen as subsemigroups of TX .Via cross-connections, Nambooripad also gave the explicitrelationship between the principal ideals of a regular semigroup.For that, he used the notion of a normal dual of a normalcategory and a cross-connection functor.

In Reg(T θX), this functor, say Γθ, is completely characterised by

the sandwich element, θ.Thus, we can realise Reg(T θ

X) as a cross-connection semigroup

(Πθ,Pθ; Γθ) = {(θa, aθ) : a ∈ Reg(T θX)}.

This representation gives the following description of thebiordered set and sandwich sets of Reg(T θ

X).

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 9 / 11

Page 43: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

In Reg(T θX), they are isomorphic to the sets P1 and P2 discussed

earlier, seen as subsemigroups of TX .Via cross-connections, Nambooripad also gave the explicitrelationship between the principal ideals of a regular semigroup.For that, he used the notion of a normal dual of a normalcategory and a cross-connection functor.In Reg(T θ

X), this functor, say Γθ, is completely characterised bythe sandwich element, θ.

Thus, we can realise Reg(T θX) as a cross-connection semigroup

(Πθ,Pθ; Γθ) = {(θa, aθ) : a ∈ Reg(T θX)}.

This representation gives the following description of thebiordered set and sandwich sets of Reg(T θ

X).

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 9 / 11

Page 44: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

In Reg(T θX), they are isomorphic to the sets P1 and P2 discussed

earlier, seen as subsemigroups of TX .Via cross-connections, Nambooripad also gave the explicitrelationship between the principal ideals of a regular semigroup.For that, he used the notion of a normal dual of a normalcategory and a cross-connection functor.In Reg(T θ

X), this functor, say Γθ, is completely characterised bythe sandwich element, θ.Thus, we can realise Reg(T θ

X) as a cross-connection semigroup

(Πθ,Pθ; Γθ) = {(θa, aθ) : a ∈ Reg(T θX)}.

This representation gives the following description of thebiordered set and sandwich sets of Reg(T θ

X).

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 9 / 11

Page 45: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

In Reg(T θX), they are isomorphic to the sets P1 and P2 discussed

earlier, seen as subsemigroups of TX .Via cross-connections, Nambooripad also gave the explicitrelationship between the principal ideals of a regular semigroup.For that, he used the notion of a normal dual of a normalcategory and a cross-connection functor.In Reg(T θ

X), this functor, say Γθ, is completely characterised bythe sandwich element, θ.Thus, we can realise Reg(T θ

X) as a cross-connection semigroup

(Πθ,Pθ; Γθ) = {(θa, aθ) : a ∈ Reg(T θX)}.

This representation gives the following description of thebiordered set and sandwich sets of Reg(T θ

X).

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 9 / 11

Page 46: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

In Reg(T θX), the idempotents are given by :

EΓθ = {(A, π) : πθ separates A and θ(A) is a cross-section of π}

They form a regular biordered set with suitably definedquasi-orders and basic products.Thus, we can describe the biordered set, completely in terms ofsubsets and partitions.Then the Sandwich set S(A, π) = S((A, π′), (A′, π)) is given by

S(A, π) = {(X,σ) : X is a cross-section of πand A is a cross-section of σ}

where A,A′, X ∈Pθ and π, π′, σ ∈ Πθ.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 10 / 11

Page 47: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

In Reg(T θX), the idempotents are given by :

EΓθ = {(A, π) : πθ separates A and θ(A) is a cross-section of π}

They form a regular biordered set with suitably definedquasi-orders and basic products.

Thus, we can describe the biordered set, completely in terms ofsubsets and partitions.Then the Sandwich set S(A, π) = S((A, π′), (A′, π)) is given by

S(A, π) = {(X,σ) : X is a cross-section of πand A is a cross-section of σ}

where A,A′, X ∈Pθ and π, π′, σ ∈ Πθ.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 10 / 11

Page 48: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

In Reg(T θX), the idempotents are given by :

EΓθ = {(A, π) : πθ separates A and θ(A) is a cross-section of π}

They form a regular biordered set with suitably definedquasi-orders and basic products.Thus, we can describe the biordered set, completely in terms ofsubsets and partitions.

Then the Sandwich set S(A, π) = S((A, π′), (A′, π)) is given by

S(A, π) = {(X,σ) : X is a cross-section of πand A is a cross-section of σ}

where A,A′, X ∈Pθ and π, π′, σ ∈ Πθ.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 10 / 11

Page 49: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

In Reg(T θX), the idempotents are given by :

EΓθ = {(A, π) : πθ separates A and θ(A) is a cross-section of π}

They form a regular biordered set with suitably definedquasi-orders and basic products.Thus, we can describe the biordered set, completely in terms ofsubsets and partitions.Then the Sandwich set S(A, π) = S((A, π′), (A′, π)) is given by

S(A, π) = {(X,σ) : X is a cross-section of πand A is a cross-section of σ}

where A,A′, X ∈Pθ and π, π′, σ ∈ Πθ.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 10 / 11

Page 50: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Further, we observe that the structural results of Dolinka andEast has a cross-connection interpretation.

For instance, the map

ψ : Reg(T θX)→ Reg(T (X,A))×Reg(T (X,α)) : a 7→ (aθ, θa)

being injective translates to the cross-connection functor being alocal isomorphism.So, this dual approach may help in extending the results to amore general class of semigroups.How we may extend this approach to the entire variantsemigroup is another question.We believe that, a solution to this problem may shed some lightinto the much more general problem of the cross-connectionconstruction of arbitrary semigroups.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 11 / 11

Page 51: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Further, we observe that the structural results of Dolinka andEast has a cross-connection interpretation.For instance, the map

ψ : Reg(T θX)→ Reg(T (X,A))×Reg(T (X,α)) : a 7→ (aθ, θa)

being injective translates to the cross-connection functor being alocal isomorphism.

So, this dual approach may help in extending the results to amore general class of semigroups.How we may extend this approach to the entire variantsemigroup is another question.We believe that, a solution to this problem may shed some lightinto the much more general problem of the cross-connectionconstruction of arbitrary semigroups.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 11 / 11

Page 52: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Further, we observe that the structural results of Dolinka andEast has a cross-connection interpretation.For instance, the map

ψ : Reg(T θX)→ Reg(T (X,A))×Reg(T (X,α)) : a 7→ (aθ, θa)

being injective translates to the cross-connection functor being alocal isomorphism.So, this dual approach may help in extending the results to amore general class of semigroups.

How we may extend this approach to the entire variantsemigroup is another question.We believe that, a solution to this problem may shed some lightinto the much more general problem of the cross-connectionconstruction of arbitrary semigroups.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 11 / 11

Page 53: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Further, we observe that the structural results of Dolinka andEast has a cross-connection interpretation.For instance, the map

ψ : Reg(T θX)→ Reg(T (X,A))×Reg(T (X,α)) : a 7→ (aθ, θa)

being injective translates to the cross-connection functor being alocal isomorphism.So, this dual approach may help in extending the results to amore general class of semigroups.How we may extend this approach to the entire variantsemigroup is another question.

We believe that, a solution to this problem may shed some lightinto the much more general problem of the cross-connectionconstruction of arbitrary semigroups.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 11 / 11

Page 54: Cross-connections and variants of TX · Variants of T XCross-connectionsSubsets and partitions Reg(T ) Biordered setSummary Cross-connections and variants of T X Azeef Muhammed P.

Variants of TX Cross-connections Subsets and partitions Reg(T θX ) Biordered set Summary

Further, we observe that the structural results of Dolinka andEast has a cross-connection interpretation.For instance, the map

ψ : Reg(T θX)→ Reg(T (X,A))×Reg(T (X,α)) : a 7→ (aθ, θa)

being injective translates to the cross-connection functor being alocal isomorphism.So, this dual approach may help in extending the results to amore general class of semigroups.How we may extend this approach to the entire variantsemigroup is another question.We believe that, a solution to this problem may shed some lightinto the much more general problem of the cross-connectionconstruction of arbitrary semigroups.

Azeef Muhammed P. A. Ural Federal University Cross-connections and variants of TX 11 / 11