Every N „-categorical distributive lattice of finite breadth has a finitely axiomatizablc theory. This result extends the analogous result for partially ordered sets of finite width. 0. Introduction. This note is mainly concerned with the following theorem. Theorem 1. Every S „-categorical distributive lattice of finite breadth has a finitely axiomatizable theory. For general references on dist...

The Weihrauch degrees and strong Weihrauch degrees are partially ordered structures representing degrees of unsolvability of various mathematical problems. Their study has been widely applied in computable analysis, complexity theory, and more recently, also in computable combinatorics. We answer an open question about the algebraic structure of the strong Weihrauch degrees, by exhibiting a joi...

This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions of distributive lattices. Each of the categories involved is generated by an object having a 2-element underlying set; additional structure may be algebraic (lattice or complete lattice operations) or relational (order) and, in either case, topology may or may not b...

The equality on the inverval I is the equality in the free bounded distributive lattice on generators i, 1− i. The equality in the face lattice F is the one for the free distributive lattice on formal generators (i = 0), (i = 1) with the relation (i = 0) ∧ (i = 1) = 0. We have [(r ∨ s) = 1] = (r = 1) ∨ (s = 1) and [(r∧s) = 1] = (r = 1)∧ (s = 1). An irreducible element of this lattice is a face,...

We apply Preuss' concept of $mbbe$-connectedness to the categories of lattice-valued uniform convergence spaces and of lattice-valued uniform spaces. A space is uniformly $mbbe$-connected if the only uniformly continuous mappings from the space to a space in the class $mbbe$ are the constant mappings. We develop the basic theory for $mbbe$-connected sets, including the product theorem. Furtherm...

Abstract. Let L be a lattice and M a bounded distributive lattice. Let ConL denote the congruence lattice of L, P (M) the Priestley dual space of M , and L (M) the lattice of continuous order-preserving maps from P (M) to L with the discrete topology. It is shown that Con(L ) ∼= (ConL) P (ConM) Λ , the lattice of continuous order-preserving maps from P (ConM) to ConL with the Lawson topology. V...

A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann, Pickering, and Roddy proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this r...

In this paper we introduce the notion of Boolean filters in a pseudo-complemented distributive lattice and characterize the class of all Boolean filters. Further a set of equivalent conditions are derived for a proper filter to become a prime Boolean filter. Also a set of equivalent conditions is derived for a pseudo-complemented distributive lattice to become a Boolean algebra. Finally, a Bool...