L-Fuzzy Congruences and L-Fuzzy Kernel Ideals in Ockham Algebras

In this paper, we study fuzzy congruence relations and kernel fuzzy ideals of an Ockham algebra A , f , whose truth values are in a complete lattice satisfying the infinite meet distributive law. Some equivalent conditions are derived for a fuzzy ideal of an Ockham algebra A to become a fuzzy kernel ideal. We also obtain the smallest (respectively, the largest) fuzzy congruence on A having a given fuzzy ideal as its kernel.

The Permutable Congruences of bpO-Algebras

An algebra A is said to be congruence permutable if any two congruences on it are permutable. This property has been investigated in several classes of algebras, for example, de Morgan algebras, p-algebras and MS-algebras, etc. In this paper, we characterize the permutable congruences on a finite Ockham algebra with balanced pseudocomplementation via the Priestley duality.

The Lattice of Congruences on a Subdirectly Irreducible Weak Stone-Ockham Algebra

Weak Stone-Ockham algebras are those algebras (L; ∧, ∨, f, ⋆,0,1) of type 〈2,2,1,1,0,0〉, where (L; ∧, ∨, f,0,1) is an Ockham algebra, (L; ∧, ∨, ⋆,0,1) is a weak Stone algebra, and the unary operations f and ⋆ commute. In this paper, we give a complete description of the structure of the lattice of congruences on the subdirectly irreducible algebras.

Ockham Algebras with Balanced Demi-pseudocomplementation

Let bdpO be the variety consisting of those algebras (L; ∧, ∨, f, *,0,1) of type 〈2,2,1,1,0,0〉, where (L; ∧, ∨, f,0,1) is an Ockham algebra, (L; ∧, ∨, *,0,1) is a demi-pseudocompleted lattice, and the unary operations f and * satisfy the conditions [f(x)]*=f2(x) and f(x*)=x**. Here we give a description of the structure of the subdirectly irreducible algebras in bdpO.

OPTIMAL NATURAL DUALITIES: THE STRUCTURE OF FAILSETS

B. A. Davey and H. A. Priestley have investigated the optimality of dualities on a quasivariety [Formula: see text], where [Formula: see text] is a finite algebra. Relative to a given set Ω of relations yielding a duality, they characterized the optimal dualities as the dualities determined by the transversals of a certain family of subsets of Ω. However the structure of these subsets — known as globally minimal failsets — remained to be understood. This paper gives a complete description of the globally minimal failsets which do not contain partial endomorphisms, and an algorithmic method to determine them. These results are applied, by way of illustration, to the variety of de Morgan algebras and to two further varieties, one of them an Ockham algebra variety and the other a variety of Heyting algebras. All the globally minimal failsets are determined in each case.

Endomorphism regular Ockham algebras of finite Boolean type

If (L; ƒ) is an Ockham algebra with dual space (X; g), then it is known that the semigroup of Ockham endomorphisms on L is (anti-)isomorphic to the semigroup Λ(X; g) of continuous order-preserving mappings on X that commute with g. Here we consider the case where L is a finite boolean lattice and ƒ is a bijection. We begin by determining the size of Λ(X;g), and obtain necessary and sufficient conditions for this semigroup to be regular or orthodox. We also describe its structure when it is a group, or an inverse semigroup that is not a group. In the former case it is a cartesian product of cyclic groups and in the latter a cartesian product of cyclic groups each with a zero adjoined.