On lattices of convex subsets of monounary algebras Z. Farkasov P. - - PowerPoint PPT Presentation

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On lattices of convex subsets of monounary algebras Z. Farkasov P. J. afrik University, Koice, Slovakia coauthor D. Jakubkov-Studenovsk Conference on Universal Algebra AAA88 Warsaw June 19-22, 2014 Introduction Structure

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On lattices of convex subsets of monounary algebras

  • Z. Farkasová
  • P. J. Šafárik University, Košice, Slovakia

coauthor D. Jakubíková-Studenovská Conference on Universal Algebra AAA88 Warsaw June 19-22, 2014

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Introduction

Structure (algebra, relational structure, topological structure, ...) there corresponds

↓

  • congruence lattice
  • quasiorder lattice
  • endomorphism monoid
  • automorphism group
  • lattice of subuniverses (substructures)
  • lattice of retracts
  • lattice of convex subsets of
  • (partially) ordered structure
  • structure with a topology
  • ordered graph
  • monounary algebra
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Introduction

Representation problems

  • group representable as an automorphism group of poset,

lattice, semilattice, unary algebra, monounary algebra, ...

  • subalgebra lattices (W. Bartol of monounary algebras)
  • lattice representable as a congruence lattice of lattices

(G. Grätzer, M. Ploščica), unary algebras (J. Berman), finite partial unary algebras (D. Jakubíková-Studenovská), monounary algebras (e.g., modular and distributive congruence lattices of monounary algebras characterized,

  • C. Ratanaprasert, S. Thiranantanakorn)
  • lattice representable as a quasiorder lattice on universal

algebras (A.G. Pinus), monounary algebras (D. Jakubíková-Studenovská)

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Introduction

Convexity

  • Geometry: convex set - natural notion, graphically visible
  • lattice of convex subsets of partially ordered sets and lattices -
  • M. K. Bennett 1977, J. Lihová 2000, M. Semenova,
  • F. Wehrung 2004
  • lattice of convex subsets of a (partial) monounary algebra

(Jakubíková-Studenovská)

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Preliminary

Definition

A monounary algebra A is a pair (A, f ) where A is a non-empty set and f : A → A is a unary operation on A.

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Preliminary

  • To a monounary algebra A = (A, f ) there corresponds a

directed graph G(A, f ) = (A, E) such that E = {(a, f (a)): a ∈ A}.

  • In this graph every vertex has outdegree 1.
  • Every graph G with outdegree 1 defines a monounary algebra
  • n its vertex set, where f (a) is the single vertex such that

(a, f (a)) is an edge in G.

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Preliminary

  • connected: ∀ x, y ∈ A ∃ n, m ∈ N ∪ {0} such that

f n(x) = f m(y)

  • connected component of (A, f ): maximal connected

subalgebra

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Preliminary

  • connected: ∀ x, y ∈ A ∃ n, m ∈ N ∪ {0} such that

f n(x) = f m(y)

  • connected component of (A, f ): maximal connected

subalgebra

  • c ∈ A is cyclic if f k(c) = c for some k ∈ N
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Preliminary

  • connected: ∀ x, y ∈ A ∃ n, m ∈ N ∪ {0} such that

f n(x) = f m(y)

  • connected component of (A, f ): maximal connected

subalgebra

  • c ∈ A is cyclic if f k(c) = c for some k ∈ N
  • the set of all cyclic elements of some connected component of

(A, f ) is a cycle of (A, f )

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Preliminary

  • connected: ∀ x, y ∈ A ∃ n, m ∈ N ∪ {0} such that

f n(x) = f m(y)

  • connected component of (A, f ): maximal connected

subalgebra

  • c ∈ A is cyclic if f k(c) = c for some k ∈ N
  • the set of all cyclic elements of some connected component of

(A, f ) is a cycle of (A, f )

  • loop - one-element cycle
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Convex subsets

Definition

A subset B ⊆ A is called convex in (A, f ) if, whenever

  • a, b, c are distinct elements of A,
  • b, c ∈ B,
  • there is an oriented path in G(A, f ) going from b to c, not

containing the element c twice and containing the element a, then a belongs to B as well.

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Convex subsets

1 3 2 4 6 5 7

Convex subsets of (A, f ):

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Convex subsets

1 3 2 4 6 5 7

Convex subsets of (A, f ):

  • ∅, {1}, . . . , {7},
  • {1, 2}, {1, 6}, {1, 7}, {5, 6},

{6, 7},

  • {1, 6, 7}, {5, 6, 7},
  • {2, 3, 4, 5},
  • {1, 2, 3, 4, 5}, {2, 3, 4, 5, 6},
  • {1, 2, 3, 4, 5, 6},

{2, 3, 4, 5, 6, 7},

  • A = {1, 2, 3, 4, 5, 6, 7}
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The lattice Co(A, f )

The system Co(A, f ) of all convex subsets of a monounary algebra (A, f ) ordered by inclusion is a lattice.

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The lattice Co(A, f )

The system Co(A, f ) of all convex subsets of a monounary algebra (A, f ) ordered by inclusion is a lattice. Let {Ki : i ∈ I} ⊆ Co(A, f ). Then

i∈I

Ki =

i∈I

Ki,

i∈I

Ki is the least convex subset of (A, f ) containing

i∈I

Ki.

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The lattice Co(A, f )

The system Co(A, f ) of all convex subsets of a monounary algebra (A, f ) ordered by inclusion is a lattice. Let {Ki : i ∈ I} ⊆ Co(A, f ). Then

i∈I

Ki =

i∈I

Ki,

i∈I

Ki is the least convex subset of (A, f ) containing

i∈I

Ki. The lattice Co(A, f ) is complete with the smallest element ∅ and the largest element A. Further, it is atomistic in the sense that each element of Co(A, f ) different from the empty set is the join

  • f some atoms. Atoms in Co(A, f ) are only all one-element subsets
  • f A.
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Basic properties of a lattice

Relation between considered lattice properties of Co(A, f ) DISTRIBUTIVE COMPLEMENTED SEMIMODULAR MODULAR SELFDUAL

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Basic properties of a lattice

Relation between considered lattice properties of Co(A, f ) DISTRIBUTIVE COMPLEMENTED ⇓ SEMIMODULAR ⇐ MODULAR SELFDUAL

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Modularity and distributivity

Theorem

A lattice L fails to be modular if and only if L contains a sublattice isomorphic to N5. A lattice L fails to be distributive if and only if L contains a sublattice isomorphic to M3 or N5.

N5 M3

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Modularity and distributivity

Theorem

Let (A, f ) be a monounary algebra. Then Co(A, f ) has a sublattice isomorphic to M3 if and only if (A, f ) contains a cycle of length greater then two.

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Modularity and distributivity

Theorem

Let (A, f ) be a monounary algebra. Then Co(A, f ) has a sublattice isomorphic to M3 if and only if (A, f ) contains a cycle of length greater then two.

Theorem

Let (A, f ) be a monounary algebra. Then Co(A, f ) contains a sublattice isomorphic to N5 if and only if there is a noncyclic element a ∈ A such that a, f (a), f 2(a) are distinct.

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Modularity and distributivity

Corollary

Let (A, f ) be a monounary algebra. The lattice Co(A, f ) is modular if and only if for each noncyclic a ∈ A, f (a) = f 2(a).

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Modularity and distributivity

Corollary

Let (A, f ) be a monounary algebra. The lattice Co(A, f ) is modular if and only if for each noncyclic a ∈ A, f (a) = f 2(a).

Theorem

Let (A, f ) be a monounary algebra. The following conditions are equivalent: (i) The lattice Co(A, f ) is distributive. (ii) If B is a connected component of (A, f ), then |B| = 2 implies that f (x) = f (y) for each x, y ∈ B. (iii) The lattice Co(A, f ) is equal to the power set P(A) of A.

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Semimodularity

Lemma

Let (A, f ) be a monounary algebra such that the lattice Co(A, f ) is

  • semimodular. Then
  • (A, f ) contains no noncyclic elements x, y such that

f (x) = f 2(x) = f (y) and f (x) = y,

  • (A, f ) contains no noncyclic elements x, y from the same

component such that f (x), f (y) are cyclic different elements.

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Semimodularity

x y x y

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Semimodularity

Theorem

Let (A, f ) be a monounary algebra. The lattice Co(A, f ) is semimodular if and only if each connected component S of (A, f ) satisfies one of the following conditions: (1) S ∼ = Z, (2) S ∼ = N or S ∼ = Nn for some n ∈ N, (3) S ∼ = Zn for some n ∈ N, (4) S ∼ = Z ∞

n

for some n ∈ N, (5) S ∼ = Z m

n or S ∼

= Z m,p

n

for some m, n, p ∈ N,

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Semimodularity

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Selfduality

Theorem

Let (A, f ) be a monounary algebra and let S be its connected

  • component. Then the lattice Co(A, f ) is selfdual if and only if

S ∼ = Zn for some n ∈ N or |{a, f (a), f 2(a)}| < 3 for each a ∈ S.

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Selfduality

Theorem

Let (A, f ) be a monounary algebra and let S be its connected

  • component. Then the lattice Co(A, f ) is selfdual if and only if

S ∼ = Zn for some n ∈ N or |{a, f (a), f 2(a)}| < 3 for each a ∈ S.

Lemma

Let (A, f ) be a monounary algebra. The following conditions are equivalent: (a) The lattice Co(A, f ) is selfdual. (b) The lattice Co(A, f ) is modular.

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Complementarity

Theorem

Let (A, f ) be a monounary algebra and let S be its connected

  • component. The lattice Co(A, f ) is complemented if and only if
  • f (x) is cyclic for each element x ∈ S,
  • if S contains a cycle C with at least two elements then either

S = C or f (x) = f (y) for some x, y ∈ S \ C.

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Complementarity

Theorem

Let (A, f ) be a monounary algebra and let S be its connected

  • component. The lattice Co(A, f ) is complemented if and only if
  • f (x) is cyclic for each element x ∈ S,
  • if S contains a cycle C with at least two elements then either

S = C or f (x) = f (y) for some x, y ∈ S \ C.

Corollary

Let (A, f ) be a monounary algebra. If the lattice Co(A, f ) is modular, then it is complemented.

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Complementarity

a

b c

∅ a b c 1 2 3 ab ac bc a1 b1 123 c3 a123 b123 c123 abc ab123 ac123 bc123 A ab1

1 2 3

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Basic properties of a lattice

Relation between considered lattice properties of Co(A, f ) DISTRIBUTIVE COMPLEMENTED SEMIMODULAR MODULAR SELFDUAL

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Basic properties of a lattice

Relation between considered lattice properties of Co(A, f ) DISTRIBUTIVE COMPLEMENTED ⇓ SEMIMODULAR ⇐ MODULAR SELFDUAL

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Basic properties of a lattice

Relation between considered lattice properties of Co(A, f ) DISTRIBUTIVE ⇒ COMPLEMENTED ⇓ ⇑ SEMIMODULAR ⇐ MODULAR ⇔ SELFDUAL

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Representation

  • partition of (A, f ) into connected components

(A, f ) =

  • i∈I

(Ai, f )

  • decomposition of the lattice Co(A, f ) into lattices

Co(A, f ) ∼ =

  • i∈I

Co(Ai, f )

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Representation

Theorem

Let L be a distributive lattice. Then L can be represented as a lattice of all convex subsets of a connected monounary algebra if and only if there exists a nonempty set A such that L ∼ = P(A).

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Representation

Theorem

Let L be a distributive lattice. Then L can be represented as a lattice of all convex subsets of a connected monounary algebra if and only if there exists a nonempty set A such that L ∼ = P(A).

Theorem

Let L be a selfdual lattice. Then L can be represented as a lattice

  • f all convex subsets of a connected monounary algebra if and only

if either L ∼ = P(A) for some nonempty set A or L ∼ = Mn for n > 2.

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Representation

Proposition

Let n ∈ N, S be an n-element set and L be a subdirect product of the lattices P(S) × Co(N). Then Co(Nn) ∼ = L if and only if each (U, V ) ∈ L satisfies one of the following conditions: (a) U = ∅ and V ∈ Co(N), (b) U = ∅ and V ∈ {∅} ∪ {[1, j]: j ∈ N} ∪ {[2, j]: j ∈ N \ {1}}.

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Representation

Proposition

Let i, m, n, p ∈ N, m, n > 1 and let Si be an i-element set. Then there exists a subdirect product (i) L1 of Mn × P(S1) such that L1 ∼ = Co(Z 1

n ),

(ii) L2 of Mn × P(S1+p) such that L2 ∼ = Co(Z 1,p

n ),

(iii) L3 of Co(Z m−1

1

) × Mn such that L3 ∼ = Co(Z m

n ),

(iv) L4 of Co(Z m−1

1

) × Mn × P(Sp) such that L4 ∼ = Co(Z m,p

n

), (v) L5 of Co(Z m

1 ) × P(Sp) such that L5 ∼

= Co(Z m,p

1

).

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Thank You for Your Attention !