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

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

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

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)

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)

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 .

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 on its vertex set, where f ( a ) is the single vertex such that ( a , f ( a )) is an edge in G .

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

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

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 )

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

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.

Convex subsets 3 4 Convex subsets of ( A , f ) : 2 5 1 6 7

Convex subsets 3 4 Convex subsets of ( A , f ) : • ∅ , { 1 } , . . . , { 7 } , • { 1 , 2 } , { 1 , 6 } , { 1 , 7 } , { 5 , 6 } , { 6 , 7 } , 2 5 • { 1 , 6 , 7 } , { 5 , 6 , 7 } , • { 2 , 3 , 4 , 5 } , • { 1 , 2 , 3 , 4 , 5 } , { 2 , 3 , 4 , 5 , 6 } , 1 6 • { 1 , 2 , 3 , 4 , 5 , 6 } , { 2 , 3 , 4 , 5 , 6 , 7 } , • A = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } 7

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.

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 { K i : i ∈ I } ⊆ Co ( A , f ) . Then K i , • � K i = � i ∈ I i ∈ I K i is the least convex subset of ( A , f ) containing � K i . • � i ∈ I i ∈ I

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 { K i : i ∈ I } ⊆ Co ( A , f ) . Then K i , • � K i = � i ∈ I i ∈ I K i is the least convex subset of ( A , f ) containing � K i . • � i ∈ I i ∈ I 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 of some atoms. Atoms in Co ( A , f ) are only all one-element subsets of A .

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

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

Modularity and distributivity Theorem A lattice L fails to be modular if and only if L contains a sublattice isomorphic to N 5 . A lattice L fails to be distributive if and only if L contains a sublattice isomorphic to M 3 or N 5 . M 3 N 5

Modularity and distributivity Theorem Let ( A , f ) be a monounary algebra. Then Co ( A , f ) has a sublattice isomorphic to M 3 if and only if ( A , f ) contains a cycle of length greater then two.

Modularity and distributivity Theorem Let ( A , f ) be a monounary algebra. Then Co ( A , f ) has a sublattice isomorphic to M 3 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 N 5 if and only if there is a noncyclic element a ∈ A such that a , f ( a ) , f 2 ( a ) are distinct.

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 ) .

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 .

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.

Semimodularity x y y x

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 ∼ = N n for some n ∈ N , (3) S ∼ = Z n for some n ∈ N , (4) S ∼ = Z ∞ for some n ∈ N , n (5) S ∼ n or S ∼ = Z m , p = Z m for some m , n , p ∈ N , n

Semimodularity

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

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 ∼ = Z n 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.

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 .

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.

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

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