Weakly Idempotent Lattices and Bilattices, Non-Idempotent Plonka Functions

Abstract

In this paper, we study weakly idempotent lattices with an additional interlaced operation. We characterize interlacity of a weakly idempotent semilattice operation, using the concept of hyperidentity and prove that a weakly idempotent bilattice with an interlaced operation is epimorphic to the superproduct with negation of two equal lattices. In the last part of the paper, we introduce the concepts of a non-idempotent Plonka function and the weakly Plonka sum and extend the main result for algebras with the well known Plonka function to the algebras with the non-idempotent Plonka function. As a consequence, we characterize the hyperidentities of the variety of weakly idempotent lattices, using non-idempotent Plonka functions, weakly Plonka sums and characterization of cardinality of the sets of operations of subdirectly irreducible algebras with hyperidentities of the variety of weakly idempotent lattices. Applications of weakly idempotent bilattices in multi-valued logic is to appear.

Keywords

• weakly idempotent semilattice
• weakly idempotent lattice
• weakly idempotent bilattice
• interlaced operation
• interlaced weakly idempotent bilattice
• hyperidentity
• non-idempotent Plonka function
• weakly Plonka sum
• weakly idempotent quasilattice