On some small varieties of distributive Ockham algebras

Abstract

J. Berman [2] initiated the study of a variety k of bounded distributive lattices endowed with a dual homomorphic operation paying particular attention to certain subvarieties km, n. Subsequently, A. Urquhart [8] named the algebras in k distributive Ockham algebras, and developed a duality theory, based on H. A. Priestley's order-topological duality for bounded distributive lattices [6], [7]. Amongst other things, Urquhart described the ordered spaces dual to the subdirectly irreducible algebras in Sif. This work was developed further still by M. S. Goldberg in his thesis and the paper [5]. Recently, T. S. Blyth and J. C. Varlet [3], in abstracting de Morgan and Stone algebras, studied a subvariety MS of the variety k1.1. The main result in [3]is that there are, up to isomorphism, nine subdirectly irreducible algebras in MS and their Hasse diagrams are exhibited. The methods employed in [3] are purely algebraic and can be generalized to show that, up to isomorphism, there are twenty subdirectly irreducible algebras in k1.1. In section 3 of this paper, we take a short cut to this result by utilizing the results of Urquhart and Goldberg. Our basic method is simple: the results of Goldberg [5] are applied to k1,1 to produce a certain eight-element algebra B1 in k1,1, whose lattice reduct is Boolean and whose subalgebras are, up to isomorphism, precisely the subdirectly irreducibles in k1.1. We then pick out of the list of twenty such algebras those belonging to the variety MS. In section 4, we sketch a purely algebraic proof along the lines followed by Blyth and Varlet in [3].