Ockham algebra

Abstract: In this paper we investigate a particular subclass of pseudocomplemented Ockham algebras $$(L; \wedge,\vee,f,^{*}, 0, 1)$$ where $$(L; \wedge,\vee, f, 0, 1)$$ is an Ockham algebra, $$(L; \wedge, \vee,^{*} , 0, 1)$$ is a p-algebra, and the operations $$x \longmapsto f(x)$$ and $$x \longmapsto x^{*}$$ satisfy the identities $$f(x^{*}) = x^{**}$$ and $$[f(x)]^{*} = f^{2}(x)$$ . We show that there are precisely eleven non-isomorphic subdirectly irreducible members in the class of these algebras and give a complete description of them.

Abstract: B. A. Davey and H. A. Priestley have investigated the optimality of dualities on a quasivariety , where 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.

Abstract: The variety pOconsists of those algebras (L;⋁,⋀,f,*,0,1) where (L;⋁,⋀,f,0,1) is an Ockham algebra, (L;⋁,⋀,f,*,0,1) is a p-algebra, and the unary operations fand *. commute. For an algebra in pK ωwe show that the compact congruences form a dual Stone lattice and use this to determine necessary and sufficient conditions for a principal congruence to be complemented. We also describe the lattice of subvarieties of pK 1,1identifying therein the biggest subvariety in which every principal congruence is complemented, and the biggest subvariety in which the intersection of two principal congruences is principal.