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},
2
5
{6, 7},
• {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 {Ki : i ∈ I } ⊆ Co(A, f ). Then
V
T
•
Ki =
Ki ,
i∈I
i∈I
W
S
•
Ki is the least convex subset of (A, f ) containing
Ki .
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 {Ki : i ∈ I } ⊆ Co(A, f ). Then
V
T
•
Ki =
Ki ,
i∈I
i∈I
W
S
•
Ki is the least convex subset of (A, f ) containing
Ki .
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
SEMIMODULAR
MODULAR
COMPLEMENTED
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 N5 .
A lattice L fails to be distributive if and only if L contains a
sublattice isomorphic to M3 or N5 .
M3
N5
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.
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.
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| =
6 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) 6= f 2 (x) = f (y ) and f (x) 6= 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 ∼
= Nn for some n ∈ N,
= N or S ∼
∼
(3) S = Zn for some n ∈ N,
(4) S ∼
= Zn∞ for some n ∈ N,
m,p
(5) S ∼
= Znm or S ∼
= Zn for some m, n, p ∈ 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∼
= Zn 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∼
= 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.
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) 6= 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) 6= 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
abc
3
1
a
ab1
ab123
ac123
bc123
a123
b123
c123
b
ab
a
ac
b
bc
a1
b1
1
c
∅
123
2
c3
3
Basic properties of a lattice
Relation between considered lattice properties of Co(A, f )
DISTRIBUTIVE
SEMIMODULAR
MODULAR
COMPLEMENTED
SELFDUAL
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
⇓
SEMIMODULAR ⇐ MODULAR
⇒
COMPLEMENTED
⇑
⇔ SELFDUAL
Representation
• partition of (A, f ) into connected components
(A, f ) =
X
(Ai , f )
i∈I
• decomposition of the lattice Co(A, f ) into lattices
Co(A, f ) ∼
=
Y
i∈I
Co(Ai , f )
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).
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
of all convex subsets of a connected monounary algebra if and only
if either L ∼
= Mn for n > 2.
= P(A) for some nonempty set A or L ∼
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 6= ∅ and V ∈ {∅} ∪ {[1, j] : j ∈ N} ∪ {[2, j] : j ∈ N \ {1}}.
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
(iii) L3 of
(iv) L4 of
(v) L5 of
1,p
Mn × P(S1+p ) such that L2 ∼
= Co(Zn ),
Co(Z1m−1 ) × Mn such that L3 ∼
= Co(Znm ),
m,p
Co(Z1m−1 ) × Mn × P(Sp ) such that L4 ∼
= Co(Zn ),
∼ Co(Z m,p ).
Co(Z m ) × P(Sp ) such that L5 =
1
1
Thank You for Your Attention !