Undecidability of Free Pseudo-Complemented Semilattlces

Abstract

Decision problem for the first order theory of free objects in equational classes of algebras was investigated for groups (Malcev [10]), semigroups (Quine [12]), commutative semigroups (Mostowski [11]), distributive lattices (Ershov [6]) and several varieties of rings (Lavrov [9]). Recently this question was solved for all varieties of Hilbert algebras and distributive pseudo-complemented lattices (see [7], [8]). In this paper we prove that the theory of all finitely generated free pseudo-complemented semilattices is undecidable. By a pseudo-complemented semi lattice (pcs for short) we mean an algebra 21 = (.A; A, -i, 0> of type such that is a meet semilattice with the smallest element 0 and the unary operation — i is defined by a/\x=Q iff x^~ -a. The class PCS of all pcs form a variety whose only non-trivial subvariety B (of Boolean algebras) is definable, relatively to PCS, by the identity An element a of a pcs is regular if — i— a — a. It is known that regular elements are exactly of the form — \b. These facts and the basic arithmetic of pcs can be found in [2]. For the main concepts in universal algebra the reader is referred to [5]. Now we recall Balbes' [1] description of finitely generated free pcs. Let n= {0, ••• , n — 1} be an arbitrary natural number. For Sen let 33 5 denote the pcs obtained from the lattice 2 of all subsets of S by adjoining a new smallest element 05. By 2(n) we mean the direct product II %$$• Sera For every subset A\J{i] of n let us define two elements of S(n) by putting Communicated by S. Takasu, December 24, 1986. Department of Logic, Jagiellonian University, Grodzka 52, 31-044 Cracow, Poland.