Ordinal and Horizontal Sums Constructing PBZ*-lattices

Abstract: PBZ*-lattices are algebraic structures related to quantum logics, which consist of bounded lattices endowed with two kinds of complements, named {\em Kleene} and {\em Brouwer}, such that the Kleene complement satisfies a weakening of the orthomodularity condition and the De Morgan laws, while the Brouwer complement only needs to satisfy the De Morgan laws for the pairs of elements with their Kleene complements. PBZ*-lattices form a variety $\mathbb{PBZL}^{\ast }$, which includes the variety $\mathbb{OML}$ of orthomodular lattices (considered with an extended signature, by letting their two complements coincide) and the variety $V(\mathbb{AOL})$ generated by the class $\mathbb{AOL}$ of antiortholattices. We investigate the congruences of antiortholattices, in particular of those obtained through certain ordinal sums and of those whose Brower complements satisfy the De Morgan laws, infer characterizations for their subdirect irreducibility and prove that even the lattice reducts of antiortholattices are directly irreducible. Since the two complements act the same on the lattice bounds in all PBZ*-lattices, we can define the horizontal sum of any nontrivial PBZ*-lattices, obtained by glueing them at their smallest and at their largest elements; a horizontal sum of two nontrivial PBZ*-lattices is a PBZ*-lattice exactly when at least one of its summands is an orthomodular lattice. We investigate the algebraic structures and the congruence lattices of these horizontal sums, then the varieties they generate. We obtain a relative axiomatization of the variety $V(\mathbb{OML}\boxplus \mathbb{AOL})$ generated by the horizontal sums of nontrivial orthomodular lattices with nontrivial antiortholattices w.r.t. $\mathbb{PBZL}^{\ast }$, as well as a relative axiomatization of the join of varieties $\mathbb{OML}\vee V(\mathbb{AOL})$ w.r.t. $V(\mathbb{OML}\boxplus \mathbb{AOL})$.