Lattice Of Equivalence Relations Research Articles

Varieties of monoids with complex lattices of subvarieties

A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. Examples of finitely universal varieties of semigroups have been available since the early 1970s, but it is unknown if there exists a finitely universal variety of monoids. The main objective of the present article is to exhibit the first examples of finitely universal varieties of monoids. The finite universality of these varieties is established by showing that the lattice of equivalence relations on every sufficiently large finite set is anti-isomorphic to some subinterval of the lattice of subvarieties.

Computability in the Lattice of Equivalence Relations

We investigate computability in the lattice of equivalence relations on the natural numbers. We mostly investigate whether the subsets of appropriately defined subrecursive equivalence relations -for example the set of all polynomial-time decidable equivalence relations- form sublattices of the lattice.

Lattices of equivalence relations closed under the operations of relation algebras

One of the longstanding problems in universal algebra is the question of which finite lattices are isomorphic to the congruence lattices of finite algebras. This question can be phrased as which finite lattices can be represented as lattices of equivalence relations on finite sets closed under certain first-order formulas. We generalize this question to a different collection of first-order formulas, giving examples to demonstrate that our new question is distinct. We then note that every lattice M n can be represented in this new way.

Approximate distributive laws and finite equational bases for finite algebras in congruence-distributive varieties

For a congruence-distributive variety, Maltsev’s construction of principal congruence relations is shown to lead to approximate distributive laws in the lattice of equivalence relations on each member. As an application, in the case of a variety generated by a finite algebra, these approximate laws yield two known results: the boundedness of the complexity of unary polynomials needed in Maltsev’s construction and the finite equational basis theorem for such a variety of finite type. An algorithmic version of the construction is included.

Join-semidistributive lattices and convex geometries

We introduce the notion of a convex geometry extending the notion of a finite closure system with the anti-exchange property known in combinatorics. This notion becomes essential for the different embedding results in the class of join-semidistributive lattices. In particular, we prove that every finite join-semidistributive lattice can be embedded into a lattice S P( A) of algebraic subsets of a suitable algebraic lattice A. This latter construction, S P( A), is a key example of a convex geometry that plays an analogous role in hierarchy of join-semidistributive lattices as a lattice of equivalence relations does in the class of modular lattices. We give numerous examples of convex geometries that emerge in different branches of mathematics from geometry to graph theory. We also discuss the introduced notion of a strong convex geometry that might promise the development of rich structural theory of convex geometries.

Distributive laws for commuting equivalence relations

In his paper (1942), Ore found necessary and sufficient conditions under which the modular and distributive laws hold in the lattice of equivalence relations on a set S. In the present paper, we consider commuting equivalence relations. It has been proved by Jónsson (1953) that the modular law holds in the lattice of commuting equivalence relations. We give some necessary and sufficient conditions for the distributive law and its dual to hold for commuting equivalence relations.

Complementation in the lattice of equivalence relations

The aim of this paper is to study those pairs of complementary equivalence relations on a fixed set which are maximal as families of mutually complementary equivalence relations. The existence of such pairs on uncountable sets was proved by Steprāns and Watson (1995). They conjectured that such pairs do not exist in the finite and in the countable case. Here we disprove this conjecture by proving that they exist in huge quantity in both cases. We study in detail the case when: (a) one of the equivalence relations in the pair has precisely two equivalence classes; (b) one of the equivalence relations has at most three equivalence classes; (c) in one of the equivalence relations all but one equivalence classes are singletons. In cases (a) and (c) we describe all pairs of complementary equivalence relations having this extremal property. In the case (a) the non-extremal pairs are related to Turan graphs.

THE IDEMPOTENTS IN A PERIODIC SEMIGROUP

Let [Formula: see text] be the semigroup variety determined by the identity xm=xm+k. For [Formula: see text] we define operations on the set E(S) of idempotents of S and thus obtain the idempotent algebra of S. For any subvariety [Formula: see text] of [Formula: see text] the idempotent algebras of the members of [Formula: see text] form a variety [Formula: see text] and [Formula: see text] yields a complete homomorphism of the lattice [Formula: see text] of subvarieties of [Formula: see text] onto the lattice [Formula: see text] of subvarieties of [Formula: see text]. The lattice [Formula: see text] contains a ∩-semilattice isomorphic to the ∩-semilattice [Formula: see text] of group varieties of exponent dividing k for every m≥1. In particular, for appropriate k, the lattice of subvarieties of the variety of all idempotent algebras of the completely regular semigroups over groups that belong to [Formula: see text] is of the power of the continuum. For any [Formula: see text], ρ→ρ|E(S) yields a complete homomorphism of the congruence lattice of S into the lattice of equivalence relations on E(S).

Mutually complementary families of 𝑇₁ topologies, equivalence relations and partial orders

We examine the maximum sizes of mutually complementary families in the lattice of topologies, the lattice of T 1 {T_1} topologies, the semi-lattice of partial orders and the lattice of equivalence relations. We show that there is a family of κ \kappa many mutually complementary partial orders (and thus T 0 {T_0} topologies) on κ \kappa and, using this family, build another family of κ \kappa many mutually T 1 {T_1} complementary topologies on κ \kappa . We obtain κ \kappa many mutually complementary equivalence relations on any infinite cardinal κ \kappa and thus obtain the simplest proof of a 1971 theorem of Anderson. We show that the maximum size of a mutually T 1 {T_1} complementary family of topologies on a set of cardinality κ \kappa may not be greater than κ \kappa unless ω > κ > 2 c \omega > \kappa > {2^c} . We show that it is consistent with and independent of the axioms of set theory that there be ℵ 2 {\aleph _2} many mutually T 1 {T_1} -complementary topologies on ω 1 {\omega _1} using the concept of a splitting sequence. We construct small maximal mutually complementary families of equivalence relations.

Lattice Theory of Generalized Partitions

In (1) the lattice of all equivalence relations on a set S was studied and many important properties were established. In (2) and (3) the lattice of all geometries on a set S was studied and it was shown to be a universal lattice which shares many properties with the lattice of equivalence relations on S. In this paper we shall give the definition of a partition of type n and investigate the lattice formed by all partitions of type n on a fixed set S. It will be seen that a partition of type one on S can be considered as an equivalence relation on S and similarly a partition of type two on S can be considered as a geometry on S as defined in (2). Thus we shall obtain a unified theory of lattices of equivalence relations, lattices of geometries and partition lattices of higher types.