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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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
).
...
.
...........................
.
........
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
).
...
.
...........................
.
........
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
).
...
.
...........................
.
........
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
).
...
.
...........................
.
........
Thank You!
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