Undecidability of free pseudocomplemented semilattices

PawełM. Idziak
1987 Publications of the Research Institute for Mathematical Sciences  
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
more » ... emented semilattices is undecidable. By a pseudo-complemented semi lattice (pcs for short) we mean an algebra 21 = (.A; A, -i, 0> of type <2, 1, 0> 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 s of all subsets of S by adjoining a new smallest element 0 5 . 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,
doi:10.2977/prims/1195176449 fatcat:j7zjypqjffg4lm4pqdjl7qhwcy