The class of continuous lattices can be characterized by infinitary equations. Therefore, it is closed under the formation of subalgebras and homomorphic images. Following the terminology of [18] we introduce a continuous lattice subframe to be a sublattice closed under the formation of arbitrary infs and directed sups. This notion corresponds with a subalgebra of a continuous lattice in [16]. ...

We show that for any tolerance R on U , the ordered sets of lower and upper rough approximations determined by R form ortholattices. These ortholattices are completely distributive, thus forming atomistic Boolean lattices, if and only if R is induced by an irredundant covering of U , and in such a case, the atoms of these Boolean lattices are described. We prove that the ordered set RS of rough...

Let Pw and PM be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty Π1 subsets of 2 , under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of Pw. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of PM .

This paper surveys recent developments in the theory of profinite Heyting algebras (resp. bounded distributive lattices, Boolean algebras) and profinite completions of Heyting algebras (resp. bounded distributive lattices, Boolean algebras). The new contributions include a necessary and sufficient condition for a profinite Heyting algebra (resp. bounded distributive lattice) to be isomorphic to...

A complete lattice L is constructively completely distributive CCD when the sup arrow from down closed subobjects of L to L has a left adjoint The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of CCD lattices and sup preserving arrows There is a restriction to order ideals and totally algebraic lattices Both biequivalences have left exact ver sions As appli...

In the late 1930's Garrett Birkhoff [3] pioneered the theory of distributive lattices. An important component in this theory is the concept of exponentiation of lattices [4]: for a lattice L and a partially ordered set P let L denote the set of all order-preserving maps of P to L partially ordered b y / ^ g if and only if/(;c) ^ g(x) for each x e P (see Figure 1). Indeed, If is a lattice. This ...

We introduce the notion of a sturdy frame of abstract algebras which is a common generalization of a sturdy semilattice of semigroups, the sum of lattice ordered systems, the strong distributive lattice of semirings, the sturdy frame of type (2, 2) algebras and the strong b-lattice of semirings. Also, we give some properties and characterizations of the sturdy frame of abstract algebras. As an ...

An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove th...

We deene the notion of weak relative pseudo-complement, and we show that, for an arbitrary lattice, the property of weak relative pseudo-complementation is strictly weaker than relative pseudo-complementation, but stronger than pseudo-complementation. Our main interest for this notion is in relation with the theory of closure operators. We prove that if a complete lattice L is completely inf-di...

Let £ be a $0$-distributive lattice with the least element $0$, the greatest element $1$, and ${rm Z}(£)$ its set of zero-divisors. In this paper, we introduce the total graph of £, denoted by ${rm T}(G (£))$. It is the graph with all elements of £ as vertices, and for distinct $x, y in £$, the vertices $x$ and $y$ are adjacent if and only if $x vee y in {rm Z}(£)$. The basic properties of the ...