Congruence lattices of semilattices (original) (raw)

The structure of congruence lattices of finite semilattices

Algebra and Logic, 1996

G. ghitomirskii gave a description of congruence lattices of semilattices in the second-order language. Here we describe finite lattices belonging to this class in terms of properties shared by coatoms (minimal nonidentity elements of a lattice}. As a consequence, a characterization of finite semiIattices with isomorphic congruence lattices is obtained. LEMMA [2]. The congruence lattice of a finite lower semilattice P is dually isomorphic to Sub+ (P). Note that if a semilattice P has a greatest element 1p, then the lattice Sub+ (P) coincides with Subv (P), the lattice of upper subsemiJattices of P with Op. Lattices of the form Subv (P) were described in , where it was shown that the property of having no special sequences of atoms, called cycles, is equivalent to being lower bounded (see also ). It was noted in [8] that lattices Sub+(P) satisfy a similar condition, and i = O, n -1. For a biatomic lattice, this definition coincides with the well-known notion of a C-cycle (see [10, ll D. For our further reasoning, we need the following: LEMMA 1.2 (see ). Let L be a finite biatomic lattice which satisfies (D2) and has no cycles and let ]P be its socle. Then the lattice S(]?) of subsystems of 1? is isomorphic to L. Specifically, if the socles 171 and ~2 of two such lattices L1 and L2 are isomorphic, then LI ~ L2.

Varieties whose congruences satisfy cerain lattice identities

Algebra Universalis, 1974

Conditions for congruence modularity were given by Day [1], and it was shown by Wille that the condition that the congruence lattices of the algebras in a variety satisfy a given lattice identity is determined by a weak Mal'cev condition [8, see also 7, 6]. In this paper we show that varieties whose congruence lattices satisfy one of" a class of lattice identities of a fairly general form (see Theorem 1) are in fact congruence modular. A similar theorem is proved for congruence distributivity. Related results appear in [2], where it is shown that a variety of semigroups whose congruences satisfy some nontrivial lattice identity is in fact a variety of periodic groups; hence, in particular, its congruence lattices are modular. Notation. Throughout .~ will represent an arbitrary variety of algebras. We will let F~ (X) denote the .q-free algebra generated by the set X, and where there is no possible ambiguity we will write F(X) for F a (X). 6)(R) will be the class of all congruence lattices O (A) for AeR. We shall use round symbols (r~, w) for set operations, and sharp symbols (^, v) for lattice operations. FL(X) will denote the free lattice generated by the set X. If weFL(X), then var(w) is the set of members of X which appear in the canonical expression of w [9]. Given a set S, let II(S) denote the partition lattice on S. If QeFI(S) and S_c U, then ~o is the member of/7(U) given by ~={(x,y):xQy in S or x=y}.

On the semidistributivity of elements in weak congruence lattices of algebras and groups

Algebra universalis, 2008

Weak congruence lattices and semidistributive congruence lattices are both recent topics in Universal Algebra. This motivates the main result of the present paper, which asserts that a finite group G is a Dedekind group if and only if the diagonal relation is a join-semidistributive element in the lattice of weak congruences of G. A variant in terms of subgroups rather than weak congruences is also given. It is pointed out that no similar result is valid for rings. An open problem and some results on the join-semidistributivity of weak congruence lattices are also included.

On the lattice of congruences on an eventually regular semigroup

Journal of the Australian Mathematical Society, 1985

A natural equivalence θ on the lattice of congruences λ(S) of a semigroup S is studied. For any eventually regular semigroup S, it is shown that θ is a congruence, each θ-class is a complete sublattice of λ(S) and the maximum element in each θ-class is determined. 1980 Mathematics subject classification (Amer. Math. Soc.): 20 M 10.

Remarks on homomorphism-homogeneous lattices and semilattices

2011

We consider lattices and semilattices enjoying the homomorphism-homogeneity property introduced recently by P. J. Cameron and J. Nešetřil. First we completely characterize all homomorphism-homogeneous lattices. Also, as a consequence of some general results, we exhibit transparent examples of semilattices both with and without this property. Finally, we show that the endomorphism monoid of , the (unique) countable universal homogeneous semilattice, embeds every finite semigroup.

Congruence lattices of pseudocomplemented semilattices

Algebra Universalis, 1979

In universal algebra, we oftentimes encounter varieties that are not especially well-behaved from any point of view, but are such that all their members have a "well-behaved core", i.e. subalgebras or quotients with satisfactory properties. Of special interest is the case in which this "core" is a retract determined by an idempotent endomorphism that is uniformly term definable (through a unary term t(x)) in every member of the given variety. Here, we try to give a unified account of this phenomenon. In particular, we investigate what happens when various congruence properties-like congruence distributivity, congruence permutability or congruence modularity-are not supposed to hold unrestrictedly in any A ∈ V, but only for congruence classes of values of the term operation t A .

The fundamental theorem of finite semidistributive lattices

Selecta Mathematica, 2021

We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff's Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form "A poset L is a finite semidistributive lattice if and only if there exists a set X with some additional structure, such that L is isomorphic to the admissible subsets of X ordered by inclusion; in this case, X and its additional structure are uniquely determined by L." The additional structure on X is a combinatorial abstraction of the notion of torsion pairs from representation theory and has geometric meaning in the case of posets of regions of hyperplane arrangements. We show how the FTFSDL clarifies many constructions in lattice theory, such as canonical join representations and passing to quotients, and how the semidistributive property interacts with other major classes of lattices. Many of our results also apply to infinite lattices.

Lattices with relative Stone congruence lattices II

2010

In this companion paper to [4] we present an alternative characterization of lattices with relative Stone congruence lattices. For the first time, the (RS)-modularity is given by a condition that, unlike its versions known so far, does not include the quantification via congruences of a lattice. It is also closer to the already known characterization in the semi-discrete case [5], [7]. We similarly present an alternative characterization of lattices whose congruence lattices satisfy the identities (En) of T. Hecht and T. Katriňák [8] which describe the subvarieties of relative Stone Heyting algebras. This second result generalizes an old characterization of G. Grätzer and E.T. Schmidt of lattices with Boolean congruence lattices [3].

A characterization of identities implying congruence modularity. I

Canadian Journal of Mathematics, 1980

0. Introduction. In his thesis and [24], J. B. Nation showed the existence of certain lattice identities, strictly weaker than the modular law, such that if all the congruence lattices of a variety of algebras J^ satisfy one of these identities, then all the congruence lattices were even modular. Moreover Freese and Jônsson showed in [10] that from this "congruence modularity" of a variety of algebras one can even deduce the (stronger) Arguesian identity.

The Lattice of Varieties of Implication Semigroups

Order

An implication semigroup is an algebra of type (2, 0) with a binary operation → and a 0-ary operation 0 satisfying the identities (x → y) → z ≈ x → (y → z), (x → y) → z ≈ [(z ′ → x) → (y → z) ′ ] ′ and 0 ′′ ≈ 0 where u ′ means u → 0 for any term u. We completely describe the lattice of varieties of implication semigroups. It turns out that this lattice is non-modular and consists of 16 elements.

The lattice of congruence lattices of algebras on a finite set

Algebra universalis

The congruence lattices of all algebras defined on a fixed finite set A ordered by inclusion form a finite atomistic lattice E. We describe the atoms and coatoms. Each meet-irreducible element of E being determined by a single unary mapping on A, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set A. Using these characterisations we deduce several properties of the lattice E; in particular, we prove that E is tolerance-simple whenever |A| ≥ 4. * supported by Slovak VEGA grant 1/0063/14 * * This research started as part of the TAMOP-4.2.1.B-10/2/KONV-2010-0001 project, supported by the European Union, co-financed by the European Social Fund 113/173/0-2.

A new property of congruence lattices of slim, planar, semimodular lattices

CGASA, 2022

The systematic study of planar semimodular lattices started in 2007 with a series of papers by G. Grätzer and E. Knapp. These lattices have connections with group theory and geometry. A planar semimodular lattice L is slim if M3 it is not a sublattice of L. In his 2016 monograph, "The Congruences of a Finite Lattice, A Proof-by-Picture Approach", the second author asked for a characterization of congruence lattices of slim, planar, semimodular lattices. In addition to distributivity, both authors have previously found specific properties of these congruence lattices. In this paper, we present a new property, the Three-pendant Three-crown Property. The proof is based on the first author's papers: 2014 (multifork extensions), 2017 (C1-diagrams), and a recent paper (lamps), introducing the tools we need.

Congruence lattices of algebras of fixed similarity type. I

Pacific Journal of Mathematics, 1979

We prove that if V is any infinite-dimensional vector space over any uncountable field F, then the congruence lattice (=subspace lattice) of V cannot be represented as a congruence lattice (of any algebra) without using at least | F \ operations. This refutes a long-standing conjecture-that one binary operation would always suffice.