In this paper, we study residuated lattices in order to give new characterizations for dense, regular and Boolean elements in residuated lattices and investigate special residuated lattices in order to obtain new characterizations for the directly indecomposable subvariety of Stonean residuated lattices. Free algebra in varieties of Stonean residuated lattices is constructed. We introduce in re...

Introduction. The elementary structure theory for finite Boolean algebras is, for our purposes, summarized in the following theorem, (see [1, p. 163], or [2, p. 344]). (*) If B is a finite Boolean algebra, then it is completely characterized by the number of its elements, a number which must be of the form 2", where n is the number of atoms of B. In this case B is faithfully represented as the ...

In this paper, we introduce the notions of $(odot, oplus)$-derivations and $(ominus, odot)$-derivations for $MV$-algebras and discuss some related results. We study the connection between these derivations on an $MV$-algebra $A$ and the derivations on its boolean center. We characterize the isotone $(odot, oplus)$-derivations and prove that $(ominus, odot)$-derivations are isotone. Finally we d...

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...

This theory defines a type constructor representing the free Boolean algebra over a set of generators. Values of type (α)formula represent propositional formulas with uninterpreted variables from type α, ordered by implication. In addition to all the standard Boolean algebra operations, the library also provides a function for building homomorphisms to any other Boolean algebra type. 1 Free Boo...

For suitable groups G we will show that one can add a Boolean algebra B by forcing in such a way that Aut(B) is almost isomorphic to G . In particular, we will give a positive answer to the following question due to J. Roitman: Is אω a possible number of automorphisms of a rich Boolean algebra? In [Ro], J. Roitman asked, is אω a possible number of automorphisms of a rich Boolean algebra? A Bool...

In order to research the logical system whose propositional value is given in a lattice from the semantic viewpoint, Xu [7] proposed the concept of lattice implication algebras, and discussed some of their properties. Xu and Qin [8] introduced the notion of implicative filters in a lattice implication algebra, and investigated some of their properties. Turunen [5] introduced the notion of Boole...

The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(N)/fin. We also discuss errors in previous results on the embeddability of the measure algebra. Introduction. The aim of this paper is to prove the following result. Main Theorem. The Open Colouring Axiom implies that the measure algebra cannot be embedded into the Boolean algebra P(N)/fin. By “the measure algeb...

We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean σ-algebra. For instance, a Boolean σ-algebra B is a measure algebra if and only if B−{0} is the union of a chain of sets C1 ⊂ C2 ⊂ ... such that for every n, (i) every antichain in Cn has at mostK(n) elements (for some integerK(n)), (ii) if {an}n is a sequence with an / ∈ Cn for each n,...

There is shown that there exists a complete, atomless, σ-centered Boolean algebra, which does not contain any regular countable subalgebra if and only if there exist a nowhere dense ultrafilter. Therefore the existence of such algebras is undecidable in ZFC. A subalgebra B of a Boolean algebra A is called regular whenever for every X ⊆ B, supB X = 1 implies supA X = 1; see e.g. Heindorf and Sha...