Involuted Semilattices and Uncertainty in Ternary Algebras

Abstract

An involuted semilattice S, ∨, − is a semilattice S, ∨ with an involution − : S → S, i.e., S, ∨, − satisfiesā = a, and a ∨ b =ā ∨b. In this paper we study the properties of such semilattices. In particular, we characterize free involuted semilattices in terms of ordered pairs of subsets of a set. An involuted semilattice S, ∨, − , 1 with greatest element 1 is said to be complemented if it satisfies a ∨ā = 1. We also characterize free complemented semilattices. We next show that complemented semilattices are related to ternary algebras. A ternary algebra T, +, * , − , 0, φ, 1 is a de Morgan algebra with a third constant φ satisfying φ =φ, and (a +ā) + φ = a +ā. If we define a third binary operation ∨ on T as a ∨ b = a * b + (a + b) * φ, then T, ∨, − , φ is a complemented semilattice.