Free almost-p-lattices
Hernando Gaitán · Czechoslovak Mathematical Journal · 1996
This work is the result of trying to describe the free almost-p-lattices going along the lines with the paper [1] of Berman and Dwinger in which the finitely generated free distributive p-lattices are described.In doing so we find that the varieties of almost-p-lattices generated by L nn , n ^ 1 (see definitions next section) are defined by the same equations used by Lee in [2] in order to describe the subvarieties of the variety of distributive p-algebras.This is accomplished in Section 4. In Section 5 we use this result to describe the join irreducible elements of a free almost p-lattices with n generators generalizing in this way to almost-p-lattices the results of Berman and Dwinger for distributive p-lattices.Section 2 is devoted to give the necessary definition and preliminares and in Section 3 some facts related to atoms of finitely generated almost-p-lattices, needed in the sequel, are studied. DEFINITIONS AND PRELIMINARIESAn almost-p-lattice (abbreviated in the sequel to apl) is an algebra (L; -f, •/ ,0,1) of type (2, 2,1,0,0) where (L; -f, •, 0,1) is a distributive lattice with greatest and least elements and the unary operation ' satisfies:• 0' = 1 and V = 0.• (x + y)' = x'y'.• (xy)" = x"y".xx' = 0.