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Page 1: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Restriction semigroups and λ-Zappa-Szép products

Rida-e Zenab

University of York

AAA94+NSAC, 15-18 June 2017

Based on joint work with Victoria Gould

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Page 2: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Contents

Definitions: Restriction Semigroups, Zappa-Szép Products andCategories

λ-Zappa-Szép products of inverse semigroups

λ-Zappa-Szép products of restriction semigroups

λ-Semidirect products of restriction semigroups

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Page 3: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Contents

Definitions: Restriction Semigroups, Zappa-Szép Products andCategories

λ-Zappa-Szép products of inverse semigroups

λ-Zappa-Szép products of restriction semigroups

λ-Semidirect products of restriction semigroups

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Page 4: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Contents

Definitions: Restriction Semigroups, Zappa-Szép Products andCategories

λ-Zappa-Szép products of inverse semigroups

λ-Zappa-Szép products of restriction semigroups

λ-Semidirect products of restriction semigroups

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Page 5: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Contents

Definitions: Restriction Semigroups, Zappa-Szép Products andCategories

λ-Zappa-Szép products of inverse semigroups

λ-Zappa-Szép products of restriction semigroups

λ-Semidirect products of restriction semigroups

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Page 6: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Restriction semigroups

Left restriction semigroups form a variety of unary semigroups, thatis, semigroups equipped with an additional unary operation, denotedby +. The identities that define a left restriction semigroup S are:

a+a = a, a+b+ = b+a+, (a+b)+ = a+b+, ab+ = (ab)+a.

We putE = {a+ : a ∈ S},

then E is a semilattice known as the semilattice of projections of S .

Dually right restriction semigroups form a variety of unarysemigroups. In this case the unary operation is denoted by ∗.

A restriction semigroup is a bi-unary semigroup S which is both leftrestriction and right restriction and which also satisfies the linkingidentities

(a+)∗ = a+ and (a∗)+ = a∗.

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Page 7: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Restriction semigroups

Left restriction semigroups form a variety of unary semigroups, thatis, semigroups equipped with an additional unary operation, denotedby +. The identities that define a left restriction semigroup S are:

a+a = a, a+b+ = b+a+, (a+b)+ = a+b+, ab+ = (ab)+a.

We putE = {a+ : a ∈ S},

then E is a semilattice known as the semilattice of projections of S .

Dually right restriction semigroups form a variety of unarysemigroups. In this case the unary operation is denoted by ∗.

A restriction semigroup is a bi-unary semigroup S which is both leftrestriction and right restriction and which also satisfies the linkingidentities

(a+)∗ = a+ and (a∗)+ = a∗.

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Page 8: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Restriction semigroups

Left restriction semigroups form a variety of unary semigroups, thatis, semigroups equipped with an additional unary operation, denotedby +. The identities that define a left restriction semigroup S are:

a+a = a, a+b+ = b+a+, (a+b)+ = a+b+, ab+ = (ab)+a.

We putE = {a+ : a ∈ S},

then E is a semilattice known as the semilattice of projections of S .

Dually right restriction semigroups form a variety of unarysemigroups. In this case the unary operation is denoted by ∗.

A restriction semigroup is a bi-unary semigroup S which is both leftrestriction and right restriction and which also satisfies the linkingidentities

(a+)∗ = a+ and (a∗)+ = a∗.

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Page 9: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Restriction semigroups

Left restriction semigroups form a variety of unary semigroups, thatis, semigroups equipped with an additional unary operation, denotedby +. The identities that define a left restriction semigroup S are:

a+a = a, a+b+ = b+a+, (a+b)+ = a+b+, ab+ = (ab)+a.

We putE = {a+ : a ∈ S},

then E is a semilattice known as the semilattice of projections of S .

Dually right restriction semigroups form a variety of unarysemigroups. In this case the unary operation is denoted by ∗.

A restriction semigroup is a bi-unary semigroup S which is both leftrestriction and right restriction and which also satisfies the linkingidentities

(a+)∗ = a+ and (a∗)+ = a∗.

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Page 10: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Zappa-Szép products of semigroups

Let S and T be semigroups and suppose that we have maps

T × S → S , (t, s) 7→ t · sT × S → T , (t, s) 7→ ts

such that for all s, s ′ ∈ S , t, t ′ ∈ T , the following hold:

(ZS1) tt ′ · s = t · (t ′ · s); (ZS3) (ts)s′= tss

′;

(ZS2) t · (ss ′) = (t · s)(ts · s ′); (ZS4) (tt ′)s = tt′·st ′s .

Define a binary operation on S × T by

(s, t)(s ′, t ′) = (s(t · s ′), ts′t ′).

Then S ×T is a semigroup, known as the Zappa-Szép product of Sand T and denoted by S ◃▹ T .

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Page 11: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Zappa-Szép products of semigroups

Let S and T be semigroups and suppose that we have maps

T × S → S , (t, s) 7→ t · sT × S → T , (t, s) 7→ ts

such that for all s, s ′ ∈ S , t, t ′ ∈ T , the following hold:

(ZS1) tt ′ · s = t · (t ′ · s); (ZS3) (ts)s′= tss

′;

(ZS2) t · (ss ′) = (t · s)(ts · s ′); (ZS4) (tt ′)s = tt′·st ′s .

Define a binary operation on S × T by

(s, t)(s ′, t ′) = (s(t · s ′), ts′t ′).

Then S ×T is a semigroup, known as the Zappa-Szép product of Sand T and denoted by S ◃▹ T .

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Page 12: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Zappa-Szép products of semigroups

Let S and T be semigroups and suppose that we have maps

T × S → S , (t, s) 7→ t · sT × S → T , (t, s) 7→ ts

such that for all s, s ′ ∈ S , t, t ′ ∈ T , the following hold:

(ZS1) tt ′ · s = t · (t ′ · s); (ZS3) (ts)s′= tss

′;

(ZS2) t · (ss ′) = (t · s)(ts · s ′); (ZS4) (tt ′)s = tt′·st ′s .

Define a binary operation on S × T by

(s, t)(s ′, t ′) = (s(t · s ′), ts′t ′).

Then S ×T is a semigroup, known as the Zappa-Szép product of Sand T and denoted by S ◃▹ T .

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Page 13: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Zappa-Szép products of monoids

If S and T are monoids then we insist that the following fouraxioms also hold:

(ZS5) t · 1S = 1S ; (ZS7) 1T · s = s;(ZS6) t1S = t; (ZS8) 1sT = 1T .

Then S ◃▹ T is a monoid with identity (1S , 1T ).

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Page 14: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Zappa-Szép products of monoids

If S and T are monoids then we insist that the following fouraxioms also hold:

(ZS5) t · 1S = 1S ; (ZS7) 1T · s = s;(ZS6) t1S = t; (ZS8) 1sT = 1T .

Then S ◃▹ T is a monoid with identity (1S , 1T ).

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Page 15: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Categories

By a category C = (C , ·,d, r), we mean a small category in thestandard sense, where · is a partial binary operation on C andd, r : C → C .

• •x

d(x) r(x)

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Page 16: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Categories

By a category C = (C , ·,d, r), we mean a small category in thestandard sense, where · is a partial binary operation on C andd, r : C → C .

• •x

d(x) r(x)

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Page 17: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Categories

By a category C = (C , ·,d, r), we mean a small category in thestandard sense, where · is a partial binary operation on C andd, r : C → C .

• • •x y

xy

d(x) r(x) = d(y)

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Page 18: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Categories

By a category C = (C , ·,d, r), we mean a small category in thestandard sense, where · is a partial binary operation on C andd, r : C → C . We say that C is a groupoid

• • •x−1

x y

xy

d(x) = xx−1 r(x) = x−1x

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Page 19: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Inductive categories

An ordered category (C , ·,d, r,≤) is a category (C , ·,d, r) witha partial order on C such that ≤ is compatible withmultiplication and if x ≤ y , then

d(x) ≤ d(y) and r(x) ≤ r(y),

and possessing restrictions and co-restrictions.

In an ordered groupoid, ≤ must be compatible with x 7→ x−1.

An inductive category (groupoid) is an ordered category(groupoid ) in which identities form a semilattice.

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Page 20: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Inductive categories

An ordered category (C , ·,d, r,≤) is a category (C , ·,d, r) witha partial order on C such that ≤ is compatible withmultiplication and if x ≤ y , then

d(x) ≤ d(y) and r(x) ≤ r(y),

and possessing restrictions and co-restrictions.In an ordered groupoid, ≤ must be compatible with x 7→ x−1.

An inductive category (groupoid) is an ordered category(groupoid ) in which identities form a semilattice.

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Page 21: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Inductive categories

An ordered category (C , ·,d, r,≤) is a category (C , ·,d, r) witha partial order on C such that ≤ is compatible withmultiplication and if x ≤ y , then

d(x) ≤ d(y) and r(x) ≤ r(y),

and possessing restrictions and co-restrictions.In an ordered groupoid, ≤ must be compatible with x 7→ x−1.

An inductive category (groupoid) is an ordered category(groupoid ) in which identities form a semilattice.

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Page 22: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

The Ehresmann-Schein-Nambooripad Theorem

The ESN TheoremThe category of inverse semigroups and homomorphisms isisomorphic to the category of inductive groupoids and inductivefunctors.

Theorem (M. V. Lawson)The category of restriction semigroups and (2,1,1)-morphisms isisomorphic to the category of inductive categories and inductivefunctors.

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Page 23: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

The Ehresmann-Schein-Nambooripad Theorem

The ESN TheoremThe category of inverse semigroups and homomorphisms isisomorphic to the category of inductive groupoids and inductivefunctors.

Theorem (M. V. Lawson)The category of restriction semigroups and (2,1,1)-morphisms isisomorphic to the category of inductive categories and inductivefunctors.

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Page 24: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

The Ehresmann-Schein-Nambooripad Theorem

The ESN TheoremThe category of inverse semigroups and homomorphisms isisomorphic to the category of inductive groupoids and inductivefunctors.

Theorem (M. V. Lawson)The category of restriction semigroups and (2,1,1)-morphisms isisomorphic to the category of inductive categories and inductivefunctors.

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Page 25: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of inverse semigroups

Theorem (Gibert and Wazzan)Let Z = S ◃▹ T be a Zappa-Szép product of inverse semigroups Sand T . Then

B◃▹(Z ) = {(a, t) ∈ S × T : tt−1 · a−1 = a−1, tt−1 · a−1a = a−1a,

(t−1)a−1a = t−1, (tt−1)a

−1a = tt−1}

is a groupoid under the restriction of the binary operation in Z .

If the action of S on T is trivial, then

Bo(S o T ) = {(a, t) ∈ S × T : tt−1 · a−1 = a−1, tt−1 · a−1a = a−1a}

is the Billhardt’s λ-semidirect of two inverse semigroups S and T .

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Page 26: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of inverse semigroups

Theorem (Gibert and Wazzan)Let Z = S ◃▹ T be a Zappa-Szép product of inverse semigroups Sand T . Then

B◃▹(Z ) = {(a, t) ∈ S × T : tt−1 · a−1 = a−1, tt−1 · a−1a = a−1a,

(t−1)a−1a = t−1, (tt−1)a

−1a = tt−1}

is a groupoid under the restriction of the binary operation in Z .

If the action of S on T is trivial, then

Bo(S o T ) = {(a, t) ∈ S × T : tt−1 · a−1 = a−1, tt−1 · a−1a = a−1a}

is the Billhardt’s λ-semidirect of two inverse semigroups S and T .

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Page 27: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of inverse semigroups

Theorem (Gibert and Wazzan)Let Z = S ◃▹ T be a Zappa-Szép product of inverse semigroups Sand T . Then

B◃▹(Z ) = {(a, t) ∈ S × T : tt−1 · a−1 = a−1, tt−1 · a−1a = a−1a,

(t−1)a−1a = t−1, (tt−1)a

−1a = tt−1}

is a groupoid under the restriction of the binary operation in Z .

If the action of S on T is trivial, then

Bo(S o T ) = {(a, t) ∈ S × T : tt−1 · a−1 = a−1, tt−1 · a−1a = a−1a}

is the Billhardt’s λ-semidirect of two inverse semigroups S and T .

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Page 28: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of inverse semigroups

Theorem (Gibert and Wazzan)Let Z = S ◃▹ T be a Zappa-Szép product of inverse semigroups Sand T . Then

B◃▹(Z ) = {(a, t) ∈ S × T : tt−1 · a−1 = a−1, tt−1 · a−1a = a−1a,

(t−1)a−1a = t−1, (tt−1)a

−1a = tt−1}

is a groupoid under the restriction of the binary operation in Z .

If the action of S on T is trivial, then

Bo(S o T ) = {(a, t) ∈ S × T : tt−1 · a−1 = a−1, tt−1 · a−1a = a−1a}

is the Billhardt’s λ-semidirect of two inverse semigroups S and T .

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Page 29: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of inverse semigroups: Special case

Theorem (Gilbert and Wazzan)Let E be a semilattice, G be a group and Z = E ◃▹ G . Supposethat (ZS7) 1 · e = e holds. Then

B◃▹(Z ) = {(e, g) ∈ E × G : (g−1)e = g−1}

is an inductive groupoid under the restriction of the binaryoperation in Z with partial order defined by the rule

(e, g) ≤ (f , h) ⇔ e ≤ f and g = hh−1·e .

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Page 30: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of inverse semigroups: Special case

Theorem (Gilbert and Wazzan)Let E be a semilattice, G be a group and Z = E ◃▹ G . Supposethat (ZS7) 1 · e = e holds. Then

B◃▹(Z ) = {(e, g) ∈ E × G : (g−1)e = g−1}

is an inductive groupoid under the restriction of the binaryoperation in Z with partial order defined by the rule

(e, g) ≤ (f , h) ⇔ e ≤ f and g = hh−1·e .

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Page 31: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of inverse semigroups: Special case

Theorem (Gilbert and Wazzan)Let E be a semilattice, G be a group and Z = E ◃▹ G . Supposethat (ZS7) 1 · e = e holds. Then

B◃▹(Z ) = {(e, g) ∈ E × G : (g−1)e = g−1}

is an inductive groupoid under the restriction of the binaryoperation in Z with partial order defined by the rule

(e, g) ≤ (f , h) ⇔ e ≤ f and g = hh−1·e .

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Page 32: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Notion of double action

Let S and T be restriction semigroups and suppose thatZ = S ◃▹ T .

We say that S and T act doubly on each other if wehave two extra maps

S × T → T , (s, t) 7→ st and S × T → S , (s, t) 7→ s ◦ t

such that for all s, s ′ ∈ S , t, t ′ ∈ T :

(1) ss′t = s( s′t); (2) s ◦ tt ′ = (s ◦ t) ◦ t ′

and actions satisfies the following compatibility conditions

(st)s = ts∗= s∗t

s(ts) = ts+= s+t.

(CP1)

and(t · s) ◦ t = s ◦ t∗ = t∗ · st · (s ◦ t) = s ◦ t+ = t+ · s

(CP2)

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Page 33: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Notion of double action

Let S and T be restriction semigroups and suppose thatZ = S ◃▹ T . We say that S and T act doubly on each other if wehave two extra maps

S × T → T , (s, t) 7→ st and S × T → S , (s, t) 7→ s ◦ t

such that for all s, s ′ ∈ S , t, t ′ ∈ T :

(1) ss′t = s( s′t); (2) s ◦ tt ′ = (s ◦ t) ◦ t ′

and actions satisfies the following compatibility conditions

(st)s = ts∗= s∗t

s(ts) = ts+= s+t.

(CP1)

and(t · s) ◦ t = s ◦ t∗ = t∗ · st · (s ◦ t) = s ◦ t+ = t+ · s

(CP2)

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Page 34: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Notion of double action

Let S and T be restriction semigroups and suppose thatZ = S ◃▹ T . We say that S and T act doubly on each other if wehave two extra maps

S × T → T , (s, t) 7→ st and S × T → S , (s, t) 7→ s ◦ t

such that for all s, s ′ ∈ S , t, t ′ ∈ T :

(1) ss′t = s( s′t); (2) s ◦ tt ′ = (s ◦ t) ◦ t ′

and actions satisfies the following compatibility conditions

(st)s = ts∗= s∗t

s(ts) = ts+= s+t.

(CP1)

and(t · s) ◦ t = s ◦ t∗ = t∗ · st · (s ◦ t) = s ◦ t+ = t+ · s

(CP2)

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Page 35: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of restriction semigroups

Let S and T be restriction semigroups and Z = S ◃▹ T . Supposethat S and T are acting doubly on each other satisfying (CP1)and (CP2).

Let

V = {(a, t) ∈ S × T : t+ · a∗ = a∗, (t+)a∗= t+, at+ · a = a, ta

∗◦t = t}.

We aim to show that V is category.

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Page 36: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of restriction semigroups

Let S and T be restriction semigroups and Z = S ◃▹ T . Supposethat S and T are acting doubly on each other satisfying (CP1)and (CP2). Let

V = {(a, t) ∈ S × T : t+ · a∗ = a∗, (t+)a∗= t+, at+ · a = a, ta

∗◦t = t}.

We aim to show that V is category.

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Page 37: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Some observations

In inverse case:

tbb−1

= t ⇒

{(t · b)−1(t · b) = tb · b−1b

(t · b)(t · b)−1 = t · bb−1

and

t−1t · b = b ⇒

{(tb)−1tb = (t−1t)b

tb(tb)−1 = (tt−1)t·b.

Reformulated to restriction case

tb+= t ⇒

{(t · b)∗ = tb · b∗

(t · b)+ = t · b+(A)

and

t∗ · b = b ⇒

{(tb)∗ = (t∗)b

(tb)+ = (t+)t·b.(B)

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Page 38: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Some observations

In inverse case:

tbb−1

= t ⇒

{(t · b)−1(t · b) = tb · b−1b

(t · b)(t · b)−1 = t · bb−1

and

t−1t · b = b ⇒

{(tb)−1tb = (t−1t)b

tb(tb)−1 = (tt−1)t·b.

Reformulated to restriction case

tb+= t ⇒

{(t · b)∗ = tb · b∗

(t · b)+ = t · b+(A)

and

t∗ · b = b ⇒

{(tb)∗ = (t∗)b

(tb)+ = (t+)t·b.(B)

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Page 39: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of restriction semigroups

Construction of a category

Theorem (V. Gould and RZ)Let S and T be restriction semigroups and suppose thatZ = S ◃▹ T is Zappa-Szép product of S and T . Suppose that theactions satisfies (CP1) and (CP2). Let

V = {(a, t) ∈ S × T : t+ · a∗ = a∗, (t+)a∗

= t+,at+ · a = a, ta

∗◦t = t}.

For (a, t) ∈ V , we suppose that, a∗ ◦ t ∈ ES and at+ ∈ ET . Alsosuppose that (A) and (B) hold. Then V is a category under therestriction of the binary operation in Z .

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Page 40: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of restriction semigroups

Construction of a category

Theorem (V. Gould and RZ)Let S and T be restriction semigroups and suppose thatZ = S ◃▹ T is Zappa-Szép product of S and T . Suppose that theactions satisfies (CP1) and (CP2).

Let

V = {(a, t) ∈ S × T : t+ · a∗ = a∗, (t+)a∗

= t+,at+ · a = a, ta

∗◦t = t}.

For (a, t) ∈ V , we suppose that, a∗ ◦ t ∈ ES and at+ ∈ ET . Alsosuppose that (A) and (B) hold. Then V is a category under therestriction of the binary operation in Z .

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Page 41: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of restriction semigroups

Construction of a category

Theorem (V. Gould and RZ)Let S and T be restriction semigroups and suppose thatZ = S ◃▹ T is Zappa-Szép product of S and T . Suppose that theactions satisfies (CP1) and (CP2). Let

V = {(a, t) ∈ S × T : t+ · a∗ = a∗, (t+)a∗

= t+,at+ · a = a, ta

∗◦t = t}.

For (a, t) ∈ V , we suppose that, a∗ ◦ t ∈ ES and at+ ∈ ET . Alsosuppose that (A) and (B) hold. Then V is a category under therestriction of the binary operation in Z .

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Page 42: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of restriction semigroups: Specialcase

We consider the Zappa-Szép product of a semilattice and a monoid.We suppose that in this case (ZS7) 1 · e = e and (ZS8)1e = 1holds. As a monoid T is reduced restriction semigroup with

t+ = 1 = t∗ for all t ∈ T ,

therefore

V = {(a, t) ∈ S × T : t+ · a∗ = a∗, (t+)a∗= t+, at+ · a = a, ta

∗◦t = t}

reduces toV ′ = {(e, t) ∈ E × T : t = te◦t}

and (CP1) and (CP2) read as

e = (t · e) ◦ t = t · (e ◦ t), (CP3)

(et)e = te = et =e (te). (CP4)

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Page 43: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of restriction semigroups: Specialcase

We consider the Zappa-Szép product of a semilattice and a monoid.We suppose that in this case (ZS7) 1 · e = e and (ZS8)1e = 1holds. As a monoid T is reduced restriction semigroup with

t+ = 1 = t∗ for all t ∈ T ,

therefore

V = {(a, t) ∈ S × T : t+ · a∗ = a∗, (t+)a∗= t+, at+ · a = a, ta

∗◦t = t}

reduces toV ′ = {(e, t) ∈ E × T : t = te◦t}

and (CP1) and (CP2) read as

e = (t · e) ◦ t = t · (e ◦ t), (CP3)

(et)e = te = et =e (te). (CP4)

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Page 44: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of restriction semigroups: Specialcase

We consider the Zappa-Szép product of a semilattice and a monoid.We suppose that in this case (ZS7) 1 · e = e and (ZS8)1e = 1holds. As a monoid T is reduced restriction semigroup with

t+ = 1 = t∗ for all t ∈ T ,

therefore

V = {(a, t) ∈ S × T : t+ · a∗ = a∗, (t+)a∗= t+, at+ · a = a, ta

∗◦t = t}

reduces toV ′ = {(e, t) ∈ E × T : t = te◦t}

and (CP1) and (CP2) read as

e = (t · e) ◦ t = t · (e ◦ t), (CP3)

(et)e = te = et =e (te). (CP4)

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Page 45: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of restriction semigroups: Specialcase

Also note that in this special case (A) and (B) are satisfiedtrivially.

Theorem (V. Gould and RZ)Let Z = E ◃▹ T be a Zappa-Szép product of a semilattice E and amonoid T . Suppose that 1 · e = e, 1e = 1 and the action of T onE and E on T satisfies (CP3) and (CP4), respectively. Alsosuppose that

e ≤ f ⇒ t f · e = t · e.

ThenV ′ = {(e, t) ∈ E × T : t = te◦t}

is an inductive category under the restriction of binary operation inZ with partial order ≤ defined by

(e, s) ≤ (f , t) if and only if e ≤ f and s = te◦t .

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Page 46: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of restriction semigroups: Specialcase

Also note that in this special case (A) and (B) are satisfiedtrivially.

Theorem (V. Gould and RZ)Let Z = E ◃▹ T be a Zappa-Szép product of a semilattice E and amonoid T . Suppose that 1 · e = e, 1e = 1 and the action of T onE and E on T satisfies (CP3) and (CP4), respectively. Alsosuppose that

e ≤ f ⇒ t f · e = t · e.

ThenV ′ = {(e, t) ∈ E × T : t = te◦t}

is an inductive category under the restriction of binary operation inZ with partial order ≤ defined by

(e, s) ≤ (f , t) if and only if e ≤ f and s = te◦t .

...

.

...........................

.

........

Page 47: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

λ-Zappa-Szép product of restriction semigroups: Specialcase

Also note that in this special case (A) and (B) are satisfiedtrivially.

Theorem (V. Gould and RZ)Let Z = E ◃▹ T be a Zappa-Szép product of a semilattice E and amonoid T . Suppose that 1 · e = e, 1e = 1 and the action of T onE and E on T satisfies (CP3) and (CP4), respectively. Alsosuppose that

e ≤ f ⇒ t f · e = t · e.

ThenV ′ = {(e, t) ∈ E × T : t = te◦t}

is an inductive category under the restriction of binary operation inZ with partial order ≤ defined by

(e, s) ≤ (f , t) if and only if e ≤ f and s = te◦t .

...

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...........................

.

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Page 48: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Obtaining a restriction semigroup

We define a pseudo product on V ′ by the rule

(e, s)⊗ (f , t) =((e, s)|r(e,s)∧d(f ,t)

)(r(e,s)∧d(f ,t)|(f , t)

)where

r(e, s) ∧ d(f , t) = (e ◦ s, 1) ∧ (f , 1) = ((e ◦ s)f , 1)

so that V ′ is a restriction semigroup with multiplication defined by

(e, s)(f , t) =(e(s · f ), s f t(e◦st)

).

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Page 49: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

When one of the action is trivial (λ-semidirect products)

Theorem (V. Gould and RZ) Let S and T be restriction semigroupsand suppose that T acts doubly on S satisfying (CP2).

LetZ = S o T and put

P = S oλ T = {(a, t) ∈ S × T : t+ · a = a}.

Then P is an inductive category with partial order ≤ defined by

(a, t) ≤ (b, u) if and only if a ≤ t+ · b , t ≤ u.

By defining a pseudo product on our inductive category we obtain arestriction semigroup P = S oλ T with multiplication defined by

(a, t)(b, u) =

(((tu)+ · a

)(t · b

), tu

).

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Page 50: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

When one of the action is trivial (λ-semidirect products)

Theorem (V. Gould and RZ) Let S and T be restriction semigroupsand suppose that T acts doubly on S satisfying (CP2). LetZ = S o T and put

P = S oλ T = {(a, t) ∈ S × T : t+ · a = a}.

Then P is an inductive category with partial order ≤ defined by

(a, t) ≤ (b, u) if and only if a ≤ t+ · b , t ≤ u.

By defining a pseudo product on our inductive category we obtain arestriction semigroup P = S oλ T with multiplication defined by

(a, t)(b, u) =

(((tu)+ · a

)(t · b

), tu

).

...

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...........................

.

........

Page 51: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

When one of the action is trivial (λ-semidirect products)

Theorem (V. Gould and RZ) Let S and T be restriction semigroupsand suppose that T acts doubly on S satisfying (CP2). LetZ = S o T and put

P = S oλ T = {(a, t) ∈ S × T : t+ · a = a}.

Then P is an inductive category with partial order ≤ defined by

(a, t) ≤ (b, u) if and only if a ≤ t+ · b , t ≤ u.

By defining a pseudo product on our inductive category we obtain arestriction semigroup P = S oλ T with multiplication defined by

(a, t)(b, u) =

(((tu)+ · a

)(t · b

), tu

).

...

.

...........................

.

........

Page 52: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

When one of the action is trivial (λ-semidirect products)

Theorem (V. Gould and RZ) Let S and T be restriction semigroupsand suppose that T acts doubly on S satisfying (CP2). LetZ = S o T and put

P = S oλ T = {(a, t) ∈ S × T : t+ · a = a}.

Then P is an inductive category with partial order ≤ defined by

(a, t) ≤ (b, u) if and only if a ≤ t+ · b , t ≤ u.

By defining a pseudo product on our inductive category we obtain arestriction semigroup P = S oλ T with multiplication defined by

(a, t)(b, u) =

(((tu)+ · a

)(t · b

), tu

).

...

.

...........................

.

........

Page 53: Restriction semigroups and -Zappa-Szép products · Restriction semigroups and -Zappa-Szép products Rida-e Zenab University of York AAA94+NSAC, 15-18 June 2017 Based on joint work

Thank You!

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