Congruence Boolean Lifting Property (original) (raw)

Functorial Properties of the Reticulation of a Universal Algebra

2021

The reticulation of an algebra A is a bounded distributive lattice whose prime spectrum of ideals (or filters), endowed with the Stone topology, is homeomorphic to the prime spectrum of congruences of A, with its own Stone topology. The reticulation allows algebraic and topological properties to be transferred between the algebra A and bounded distributive lattices, a transfer which is facilitated if we can define a reticulation functor from a variety containing A to the variety of (bounded) distributive lattices. In this paper, we continue the study of the reticulation of a universal algebra initiated in [24], where we have used the notion of a prime congruence introduced through the term condition commutator. We characterize morphisms which admit an image through the reticulation and investigate the kinds of varieties that admit reticulation functors; we prove that these include semi-degenerate congruence-distributive varieties with the Compact Intersection Property and semi-degenerate congruence-distributive varieties with congruence intersection terms, as well as generalizations of these, and additional varietal properties ensure that the reticulation functors preserve the injectivity of morphisms. We also study the property of morphisms of having an image through the reticulation in relation to another property, involving the complemented elements of congruence lattices, exemplify the transfer of properties through the reticulation with conditions Going Up, Going Down, Lying Over and the Congruence Boolean Lifting Property, and illustrate the applicability of such a transfer by using it to derive results for certain types of varieties from properties of bounded distributive lattices.

When a quotient of a distributive lattice is a boolean algebra

arXiv: Rings and Algebras, 2019

In this article, we introduce a lattice congruence with respect to a nonempty ideal III of a distributive lattice LLL and a derivation ddd on LLL denoted by thetaId\theta_I^dthetaId. We investigate some necessary and sufficient conditions for the quotient algebra L/thetaIdL/\theta_I^dL/thetaId to become a Boolean algebra.

On Ockham algebras: Congruence lattices and subdirectly irreducible algebras

Studia Logica, 1995

Distributive bounded lattices with a dual homomorphism as unary operation, called Ockham algebras, were firstly studied by Berman (1977). The varieties of Boolean algebras, De Morgan algebras, Kleene algebras and Stone algebras are some of the well known subvarieties of Ockham algebra. In this paper, new results about the congruence lattice of Ockham algebras are given. From these results and Urquhart's representation theorem for Ockham algebras a complete characterization of the subdirectly irreducible Ockham algebras is obtained. These results are particularized for a large number of subvarieties of Ockham algebras. For these subvarieties a full description of their subdirectly irreducible algebras is given as well.

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.

The congruence lattice of implication algebras

The variety of implicat.ion algebras is a minimal quasi variety. It is 3-filtral but not 2-filtral. An implication algebra A is tolerance-trivial iff (A,:S) is a lattice, where the partial ordering I':S" is defined as follows: a :S b {:} 3x E A such that b = x. a.

Weak distributive laws and their role in lattices of congruences and equational theories

Algebra Universalis, 1988

By a result of Pigozzi and Kogalovskil, every algebraic lattice L having a completely join-irreducible top element can be represented as the lattice L(~') of equational theories extending some fixed theory Z. Conversely, strengthening a recent result due to Lampe, we show that such a representation L ~ L(-y) forces L to satisfy the following condition: if the top element of L is the join of a nonempty subset B of L then there are elements b~ ..... b,, s B such that a = (-(((b~ a a) v b2) A a).-v b~) A a for all a E L. In presence of modularity, this equation reduces to the identity a = (a A b~) v " " ~ v (a ^ b,). Motivated by these facts, we study several weak forms of distributive laws in arbitrary lattices and related types of prime elements. The main tool for applications to universal algebra is a generalized version of Lampe's Zipper Lemma. top element 1 is not only compact but completely join-irreducible (i.e. ~3 ve B _c L and I=VB imply 1 e B) is isomorphic to some L(Z), and KogalovskH announced the same theorem at the Szeged Conference 1983. Pigozzi's paper will appear in Acta Sci. Math. (Szeged). Recently, Lampe [8] has discovered a certain purely algebraic property that necessarily holds in any lattice L isomorphic to some L(Z), namely: (~) For alla, c6LandBc_L, ifVB=l andaAb=cforallbeBthena=c. Lampe's original proof uses McKenzie's Lemma stating that L(Z) is isomorphic to the congruence lattice of some algebra A having a binary term function with a Presented by Ralph Freese.

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.

Congruence Preservation, Lattices and Recognizability

ArXiv, 2020

Looking at some monoids and (semi)rings (natural numbers, integers and p-adic integers), and more generally, residually finite algebras (in a strong sense), we prove the equivalence of two ways for a function on such an algebra to behave like the operations of the algebra. The first way is to preserve congruences or stable preorders. The second way is to demand that preimages of recognizable sets belong to the lattice or the Boolean algebra generated by the preimages of recognizable sets by derived unary operation of the algebra (such as translations, quotients,. . . ).