On Regular Congruences of Partially Ordered
Semigroups
Boža Tasić
Ryerson University, Toronto
June 21, 2015
Partially Ordered Semigroups
I
A partially ordered semigroup (po-semigroup) is an ordered
triple S = hS, ·, ≤i such that
Partially Ordered Semigroups
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A partially ordered semigroup (po-semigroup) is an ordered
triple S = hS, ·, ≤i such that
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a) hS, ·, i is a semigroup,
Partially Ordered Semigroups
I
A partially ordered semigroup (po-semigroup) is an ordered
triple S = hS, ·, ≤i such that
I
I
a) hS, ·, i is a semigroup,
b) hS, ≤i is a partially ordered set (poset),
Partially Ordered Semigroups
I
A partially ordered semigroup (po-semigroup) is an ordered
triple S = hS, ·, ≤i such that
I
I
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a) hS, ·, i is a semigroup,
b) hS, ≤i is a partially ordered set (poset),
c) (x ≤ y & u ≤ v ) → x · u ≤ y · v for all x, y , u, v ∈ S.
Partially Ordered Semigroups
I
A partially ordered semigroup (po-semigroup) is an ordered
triple S = hS, ·, ≤i such that
I
I
I
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a) hS, ·, i is a semigroup,
b) hS, ≤i is a partially ordered set (poset),
c) (x ≤ y & u ≤ v ) → x · u ≤ y · v for all x, y , u, v ∈ S.
Condition c) is equivalent to
Partially Ordered Semigroups
I
A partially ordered semigroup (po-semigroup) is an ordered
triple S = hS, ·, ≤i such that
I
I
I
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a) hS, ·, i is a semigroup,
b) hS, ≤i is a partially ordered set (poset),
c) (x ≤ y & u ≤ v ) → x · u ≤ y · v for all x, y , u, v ∈ S.
Condition c) is equivalent to
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c’) x ≤ y → (x · z ≤ y · z & z · x ≤ z · y ) for all x, y , z ∈ S
Congruence Relations of Semigroups
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A congruence relation of the semigroup S = hS, ·i is an
equivalence relation θ ⊆ S 2 that satisfies the following
compatibility property:
Congruence Relations of Semigroups
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A congruence relation of the semigroup S = hS, ·i is an
equivalence relation θ ⊆ S 2 that satisfies the following
compatibility property:
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(x θ y & u θ v ) → x · u θ y · v
for all x, y , u, v ∈ S.
(CP)
Congruence Relations of Semigroups
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Given a congruence relation θ of the semigroup S = hS, ·i we
define the quotient semigroup (a.k.a the homomorphic image)
of S by θ to be the semigroup S/θ = hS/θ, i where
Congruence Relations of Semigroups
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Given a congruence relation θ of the semigroup S = hS, ·i we
define the quotient semigroup (a.k.a the homomorphic image)
of S by θ to be the semigroup S/θ = hS/θ, i where
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x/θ y /θ = (x · y )/θ
for all x, y ∈ S.
Congruence Relations of Semigroups
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Given a congruence relation θ of the semigroup S = hS, ·i we
define the quotient semigroup (a.k.a the homomorphic image)
of S by θ to be the semigroup S/θ = hS/θ, i where
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x/θ y /θ = (x · y )/θ
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The set of all congruences of a semigroup S = hS, ·i is
denoted by Con(S)
for all x, y ∈ S.
Congruence Relations of Semigroups
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How does one extend the concept of quotients to
po-semigroups?
Congruence Relations of Semigroups
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How does one extend the concept of quotients to
po-semigroups?
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For structures that have only operations, i.e., algebras,
quotients are defined using congruences.
Congruence Relations of Semigroups
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How does one extend the concept of quotients to
po-semigroups?
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For structures that have only operations, i.e., algebras,
quotients are defined using congruences.
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Structures that have both operations and relations are studied
in model theory and their quotients are defined as
The algebraic quotient + Relations on the algebraic quotient.
Quotients of Po-Semigroups
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Given a po-semigroup S = hS, ·, ≤i and θ ∈ ConhS, ·i we
define the quotient po-semigroup (a.k.a the homomorphic
image) of S by θ to be the structure S/θ = hS/θ, , i,
where
Quotients of Po-Semigroups
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Given a po-semigroup S = hS, ·, ≤i and θ ∈ ConhS, ·i we
define the quotient po-semigroup (a.k.a the homomorphic
image) of S by θ to be the structure S/θ = hS/θ, , i,
where
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a) hS/θ, i is a quotient semigroup,
Quotients of Po-Semigroups
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Given a po-semigroup S = hS, ·, ≤i and θ ∈ ConhS, ·i we
define the quotient po-semigroup (a.k.a the homomorphic
image) of S by θ to be the structure S/θ = hS/θ, , i,
where
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a) hS/θ, i is a quotient semigroup,
b) The relation on S/θ is defined by
x/θ y /θ ↔ (∃x1 ∈ x/θ)(∃y1 ∈ y /θ) x1 ≤ y1
Quotients of Po-Semigroups
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Quotients preserve identities satisfied by the original algebra
Quotients of Po-Semigroups
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Quotients preserve identities satisfied by the original algebra
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Quotients may not preserve all formulas involving relations
satisfied by the original structure.
Quotients of Po-Semigroups
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Quotients preserve identities satisfied by the original algebra
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Quotients may not preserve all formulas involving relations
satisfied by the original structure.
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A quotient of a semigroup is a semigroup,
Quotients of Po-Semigroups
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Quotients preserve identities satisfied by the original algebra
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Quotients may not preserve all formulas involving relations
satisfied by the original structure.
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A quotient of a semigroup is a semigroup,
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Unfortunately, a quotient of a po-semigroup is not necessarily
a po-semigroup.
Quotients of Po-Semigroups
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The quotient structure S/θ = hS/θ, , i satisfies
Quotients of Po-Semigroups
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The quotient structure S/θ = hS/θ, , i satisfies
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∀x (x x),
Quotients of Po-Semigroups
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The quotient structure S/θ = hS/θ, , i satisfies
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∀x (x x),
and on S/θ is compatible with , i.e.,
(x/θ y /θ ∧ u/θ v /θ) → x/θ u/θ y /θ v /θ (CPQ)
Quotients of Po-Semigroups
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The quotient structure S/θ = hS/θ, , i satisfies
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∀x (x x),
and on S/θ is compatible with , i.e.,
(x/θ y /θ ∧ u/θ v /θ) → x/θ u/θ y /θ v /θ (CPQ)
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Neither anti-symmetry nor transitivity is preserved by in
general.
Regular Congruences
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Let S = hS, ·, ≤i be a po-semigroup and θ ∈ ConhS, ·i. The
congruence θ is called regular if there exists an order on
S/θ such that:
Regular Congruences
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Let S = hS, ·, ≤i be a po-semigroup and θ ∈ ConhS, ·i. The
congruence θ is called regular if there exists an order on
S/θ such that:
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a) S/θ = hS/θ, , i is a po-semigroup,
Regular Congruences
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Let S = hS, ·, ≤i be a po-semigroup and θ ∈ ConhS, ·i. The
congruence θ is called regular if there exists an order on
S/θ such that:
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a) S/θ = hS/θ, , i is a po-semigroup,
b) The mapping ϕ : S 7→ S/θ defined by ϕ(x) = x/θ is
isotone, i.e.,
x ≤ y → x/θ y /θ.
Regular Congruences
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Let S = hS, ·, ≤i be a po-semigroup and θ ∈ ConhS, ·i. The
congruence θ is called regular if there exists an order on
S/θ such that:
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I
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a) S/θ = hS/θ, , i is a po-semigroup,
b) The mapping ϕ : S 7→ S/θ defined by ϕ(x) = x/θ is
isotone, i.e.,
x ≤ y → x/θ y /θ.
Regularity can be expressed using a special property of θ !
Open Bracelet Modulo θ
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Let S = hS, ·, ≤i be a po-semigroup and θ ∈ ConhS, ·i. A
sequence of elements (a1 , a2 , a3 , . . . , a2n ) in S is called
Open Bracelet Modulo θ
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Let S = hS, ·, ≤i be a po-semigroup and θ ∈ ConhS, ·i. A
sequence of elements (a1 , a2 , a3 , . . . , a2n ) in S is called
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an open bracelet modulo θ if it satisfies
a1 θ a2 ≤ a3 θ a4 ≤ . . . · · · ≤ a2n−1 θ a2n
Open Bracelet Modulo θ
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Let S = hS, ·, ≤i be a po-semigroup and θ ∈ ConhS, ·i. A
sequence of elements (a1 , a2 , a3 , . . . , a2n ) in S is called
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an open bracelet modulo θ if it satisfies
a1 θ a2 ≤ a3 θ a4 ≤ . . . · · · ≤ a2n−1 θ a2n
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a closed bracelet modulo θ if it satisfies
a1 θ a2 ≤ a3 θ a4 ≤ . . . · · · ≤ a2n−1 θ a2n ≤ a1
Open Bracelet Modulo θ
a1
•
a3
•
θ
•
a2
θ
•
•
......
a2n−1
•
...... θ
•
a4
Figure : Open bracelet modulo θ
θ
•
a2n
Closed Bracelet Modulo θ
a1
•
a3
•
θ
•
a2
θ
•
a2n−1
•
θ
•
θ
•
a4
Figure : Closed bracelet modulo θ
θ
•
a2n
Regular Congruences
Theorem
Let S = hS, ·, ≤i be a po-semigroup. A congruence θ ∈ ConhS, ·i
is regular if and only if it satisfies the property: for every n ∈ N
and every a1 , . . . , a2n ∈ S
a1 θ a2 ≤ a3 θ a4 ≤ . . . . . . a2n−1 θ a2n ≤ a1 → {a1 , . . . , a2n }2 ⊆ θ.
Regular Congruences
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A congruence θ ∈ ConhS, ·i is regular if and only if every
closed bracelet modulo θ is contained in a single equivalence
class modulo θ.
Regular Congruences
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A congruence θ ∈ ConhS, ·i is regular if and only if every
closed bracelet modulo θ is contained in a single equivalence
class modulo θ.
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If θ ∈ ConhS, ·i is regular then θ-classes are convex subsets of
hS, ≤i.
Lattice of Regular Congruences
Theorem
For a given po-semigroup S = hS, ·, ≤i, the set of all regular
congruences is a complete lattice with respect to the inclusion as
the ordering. We denote it by RCon S = hRConS, ⊆i.
Lattice of Regular Congruences
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The lattice RCon S is not a sublattice of the lattice Con S.
Lattice of Regular Congruences
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The lattice RCon S is not a sublattice of the lattice Con S.
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The intersection of two regular congruences is a regular
congruence.
Lattice of Regular Congruences
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The lattice RCon S is not a sublattice of the lattice Con S.
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The intersection of two regular congruences is a regular
congruence.
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The join of two regular congruences is not a regular
congruence in general.
Lattice of Regular Congruences
Theorem (Tasic)
For a po-semigroup S = hS, ·, ≤i, there is an algebraic closure
operator ΘR on S × S such that the closed subsets of S × S are
precisely the regular congruences on S. Hence
RCon S = hRCon(S), ⊆i is an algebraic lattice.
Lattice of Regular Congruences
Proof.
Consider the algebra
S∗ = hS × S, , (a, a)a∈S , s, t, (eijn )n ≥ 2, 1 ≤ i < j ≤ 2n i
where (a, a)a∈S are constants (nullary operations), s is a unary
operation, and t are binary operations, and for each n ≥ 2 and
1 ≤ i < j ≤ 2n, eijn is an n-ary operation.
Lattice of Regular Congruences
Proof.
The operations are defined by
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(a, b) (c, d) = (a · c, b · d)
Lattice of Regular Congruences
Proof.
The operations are defined by
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(a, b) (c, d) = (a · c, b · d)
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s((a, b)) = (b, a)
Lattice of Regular Congruences
Proof.
The operations are defined by
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(a, b) (c, d) = (a · c, b · d)
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s((a, b)) = (b, a)
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(
(a, d),
t((a, b) (c, d)) =
(a, b),
if b = c
otherwise
Lattice of Regular Congruences
Proof.
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eijn ((a1 , a2 ), (a3 , a4 ), . . . , (a2n−1 , a2n ))
(
(ai , aj ), if for all k=1,. . . ,n=
(a1 , a2 ), otherwise.
Lattice of Regular Congruences
Proof.
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eijn ((a1 , a2 ), (a3 , a4 ), . . . , (a2n−1 , a2n ))
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(
(ai , aj ), if for all k=1,. . . ,n=
(a1 , a2 ), otherwise.
(ai , aj ), if for all k=1,. . . ,n-1 a2k ≤ a2k+1 and a2n ≤ a1
Lattice of Regular Congruences
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Let ΘR be the Sg closure operator on S × S for the algebra
S∗ .
Lattice of Regular Congruences
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Let ΘR be the Sg closure operator on S × S for the algebra
S∗ .
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The compact members of RCon S are the finitely generated
members ΘR ((a1 , b1 ), (a2 , b2 ), . . . (an , bn )) of RCon S.
Regular Congruences Generated by a Set
For S = hS, ·, ≤i a po-semigroup and a1 , a2 , . . . an ∈ A let
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ΘR (a1 , a2 , . . . an ) denote the regular congruence generated by
{(ai , aj ) | 1 ≤ i, j ≤ n}
Regular Congruences Generated by a Set
For S = hS, ·, ≤i a po-semigroup and a1 , a2 , . . . an ∈ A let
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ΘR (a1 , a2 , . . . an ) denote the regular congruence generated by
{(ai , aj ) | 1 ≤ i, j ≤ n}
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The congruence ΘR (a1 , a2 ) is called a principal regular
congruence.
Regular Congruences Generated by a Set
For S = hS, ·, ≤i a po-semigroup and a1 , a2 , . . . an ∈ A let
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ΘR (a1 , a2 , . . . an ) denote the regular congruence generated by
{(ai , aj ) | 1 ≤ i, j ≤ n}
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The congruence ΘR (a1 , a2 ) is called a principal regular
congruence.
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For an arbitrary set X ⊆ S, let ΘR (X ) denote the regular
congruence generated by X × X .
Regular Congruences Generated by a Set
Lemma (Tasic)
Let S = hS, ·, ≤i be a po-semigroup, and suppose that
a1 , b1 , . . . an , bn ∈ S and θ ∈ RCon S. Then
Regular Congruences Generated by a Set
Lemma (Tasic)
Let S = hS, ·, ≤i be a po-semigroup, and suppose that
a1 , b1 , . . . an , bn ∈ S and θ ∈ RCon S. Then
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a) ΘR (a1 , b1 ) = ΘR (b1 , a1 )
Regular Congruences Generated by a Set
Lemma (Tasic)
Let S = hS, ·, ≤i be a po-semigroup, and suppose that
a1 , b1 , . . . an , bn ∈ S and θ ∈ RCon S. Then
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a) ΘR (a1 , b1 ) = ΘR (b1 , a1 )
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b) ΘR ((a1 , b1 ), . . . (an , bn )) = ΘR (a1 , b1 ) ∨ · · · ∨ ΘR (an , bn )
Regular Congruences Generated by a Set
Lemma (Tasic)
Let S = hS, ·, ≤i be a po-semigroup, and suppose that
a1 , b1 , . . . an , bn ∈ S and θ ∈ RCon S. Then
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a) ΘR (a1 , b1 ) = ΘR (b1 , a1 )
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b) ΘR ((a1 , b1 ), . . . (an , bn )) = ΘR (a1 , b1 ) ∨ · · · ∨ ΘR (an , bn )
S
W
c) θ = {ΘR (a, b) | (a, b) ∈ θ} = {ΘR (a, b) | (a, b) ∈ θ}.
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Lattice of Regular Congruences
Given a nonempty set of regular congruences θi for i ∈ I we have
V
T
I
R {θi | i ∈ I } =
i∈I θi
Lattice of Regular Congruences
Given a nonempty set of regular congruences θi for i ∈ I we have
V
T
I
θi
R {θi | i ∈ I } =
W i∈I
I What is the join
R in RCon S?
Lattice of Regular Congruences
Given a nonempty set of regular congruences θi for i ∈ I we have
V
T
I
θi
R {θi | i ∈ I } =
W i∈I
I What is the join
R in RCon S?
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In general, the join of two regular congruences is not a regular
congruence
Lattice of Regular Congruences
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Given θ ∈ Con S, we define the regular closure of θ, denoted
by θc , to be the smallest regular congruence containing θ.
Lattice of Regular Congruences
I
I
Given θ ∈ Con S, we define the regular closure of θ, denoted
by θc , to be the smallest regular congruence containing θ.
T
We have θc = {σ | σ ∈ RConS and θ ⊆ σ}.
Lattice of Regular Congruences
Given a nonempty set of regular congruences θi for i ∈ I we have
c
W
W
I
R {θi | i ∈ I } =
i∈I θi
Lattice of Regular Congruences
Theorem
Let S = hS, ·, ≤i be a po-semigroup and let θ ∈ Con S. The
regular closure θc of θ is given by
x θc y ↔ there is a closed bracelet modulo θ containing x, y .
Lattice of Regular Congruences
Theorem
Let S = hS,
W ·, ≤i be a po-semigroup and let θi ∈ Con S for i ∈ I .
The
join
R in RCon S is given by
W
W
x R {θi | i ∈ I } y ↔ there is a closed bracelet modulo i∈I θi
containing x, y .
Example
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Let S = h{1, a, b, c} ·, ≤i be a po-semigroup defined by the
following multiplication table and the ordering.
Example
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Let S = h{1, a, b, c} ·, ≤i be a po-semigroup defined by the
following multiplication table and the ordering.
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·
1
a
b
c
1
1
a
b
c
a
a
a
c
c
b
b
c
b
c
c
c
c
c
c
c
b
a
1
Example
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The po-semigroup S has seven congruences determined by the
following partitions:
Example
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The po-semigroup S has seven congruences determined by the
following partitions:
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0S = {{1}, {a}, {b}, {c}}
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θ1 = {{1}, {a}, {b, c}}
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θ2 = {{1}, {a, c}, {b}}
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θ3 = {{1, b}, {a, c}}
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θ4 = {{1}, {a, b, c}}
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θ5 = {{1, a}, {b, c}}
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1S = {{1, a, b, c}}
Example
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The lattice of congruences of S is
Example
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The lattice of congruences of S is
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1S
•
θ5
θ4
•
θ3
•
θ1
•
θ2
•
•
0S
•
Example
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We have
c θi a ≤ b θi b ≤ c θ i c
but
{a, b, c}2 6⊆ θi
for
i = 2, 3
Example
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We have
c θi a ≤ b θi b ≤ c θ i c
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but
{a, b, c}2 6⊆ θi
So, θ2 and θ3 are not regular congruences.
for
i = 2, 3
Example
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We have five regular congruences of S and the corresponding
lattice of regular congruences is
Example
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We have five regular congruences of S and the corresponding
lattice of regular congruences is
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0S = {{1}, {a}, {b}, {c}}
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θ1 = {{1}, {a}, {b, c}}
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θ4 = {{1}, {a, b, c}}
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θ5 = {{1, a}, {b, c}}
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1S = {{1, a, b, c}}
1S
θ5
θ4
θ1
0S
Lattice of Regular Congruences
Lemma (Tasic)
If S = hS, ·, ≤i is a po-semigroup, then for any a, b ∈ S we have
ΘR (a, b) = (Θ(a, b))c
Translations of a semigroup
Given a po-semigroup S = hS, ·, ≤i
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0-translation of S is a function g : S → S such that g (x) = x
or g (x) = a for every x ∈ S and a fixed element a ∈ S.
Translations of a semigroup
Given a po-semigroup S = hS, ·, ≤i
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0-translation of S is a function g : S → S such that g (x) = x
or g (x) = a for every x ∈ S and a fixed element a ∈ S.
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1-translation of S is a 0-translation or a function g : S → S
such that g (x) = a · x or g (x) = x · a for every x ∈ S and a
fixed element a ∈ S.
Translations of a semigroup
Given a po-semigroup S = hS, ·, ≤i
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0-translation of S is a function g : S → S such that g (x) = x
or g (x) = a for every x ∈ S and a fixed element a ∈ S.
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1-translation of S is a 0-translation or a function g : S → S
such that g (x) = a · x or g (x) = x · a for every x ∈ S and a
fixed element a ∈ S.
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A composition of k 1-translations of S is called a k-translation
of S.
Translations of a semigroup
Given a po-semigroup S = hS, ·, ≤i
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0-translation of S is a function g : S → S such that g (x) = x
or g (x) = a for every x ∈ S and a fixed element a ∈ S.
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1-translation of S is a 0-translation or a function g : S → S
such that g (x) = a · x or g (x) = x · a for every x ∈ S and a
fixed element a ∈ S.
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A composition of k 1-translations of S is called a k-translation
of S.
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A function g : S → S is called a translation of S if g is a
k-translation of S for some k.
Congruence Generation Theorem
Theorem
If S = hS, ·, ≤i is a po-semigroup, then for any a, b ∈ S we have
(c, d) ∈ Θ(a, b) iff there exist translations g0 , g1 , . . . , gn of S such
that g1 (a) = g0 (b), . . . , gn (a) = gn−1 (b) and c = g0 (a), d = gn (b)
or d = g0 (a), c = gn (b).
Open bracelet modulo θ = Θ(a, b)
c=g0 (a)
•
g1 (a)
•
θ
θ
g2 (a)
gn (a)
•
......
•
...... θ
θ
•
•
•
•
g0 (b)
g1 (b)
gn−1 (b)
gn (b)=d
Figure : Open bracelet modulo θ = Θ(a, b)
Regular Congruence Generation Theorem
Theorem (Tasic)
If S = hS, ·, ≤i is a po-semigroup, then for any a, b, c, d ∈ S we
have (c, d) ∈ ΘR (a, b) iff there exist n ∈ N, e1 , e2 , . . . , e2n ∈ S
and translations gj2i−1 for i = 1, 2, . . . , n and j = 0, 1, . . . , mi such
that
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c, d ∈ {e1 , e2 , . . . , e2n }
Regular Congruence Generation Theorem
Theorem (Tasic)
If S = hS, ·, ≤i is a po-semigroup, then for any a, b, c, d ∈ S we
have (c, d) ∈ ΘR (a, b) iff there exist n ∈ N, e1 , e2 , . . . , e2n ∈ S
and translations gj2i−1 for i = 1, 2, . . . , n and j = 0, 1, . . . , mi such
that
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c, d ∈ {e1 , e2 , . . . , e2n }
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For every i = 1, 2, . . . , n and j = 0, 1, . . . , mi
e2i−1 = g02i−1 (a)
2i−1
gj2i−1 (b) = gj−1
(a)
2i−1
gm
(b) = e2i
i
e2i = g02i−1 (a)
or
2i−1
gj2i−1 (b) = gj−1
(a)
2i−1
gm
(b) = e2i−1
i
Regular Congruence Generation Theorem
Theorem (Tasic)
If S = hS, ·, ≤i is a po-semigroup, then for any a, b, c, d ∈ S we
have (c, d) ∈ ΘR (a, b) iff there exist n ∈ N, e1 , e2 , . . . , e2n ∈ S
and translations gj2i−1 for i = 1, 2, . . . , n and j = 0, 1, . . . , mi such
that
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c, d ∈ {e1 , e2 , . . . , e2n }
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For every i = 1, 2, . . . , n and j = 0, 1, . . . , mi
e2i−1 = g02i−1 (a)
2i−1
gj2i−1 (b) = gj−1
(a)
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e2i = g02i−1 (a)
or
2i−1
gj2i−1 (b) = gj−1
(a)
2i−1
2i−1
gm
(b) = e2i
gm
(b) = e2i−1
i
i
For every i = 1, 2, . . . , n − 1, e2i ≤ e2i+1 and e2n ≤ e1 .