INVOLUTED SEMILATTICES AND UNCERTAINTY IN TERNARY ALGEBRAS

摘要

An involuted semilattice is a semilattice with an involution -: S→S, i.e., satisfies a-bar = a, and (a∨b)-bar = a-bar∨b-bar. 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 with greatest element 1 is said to be complemented if it satisfies a∨a-bar=1. We also characterize free complemented semilattices. We next show that complemented semilattices are related to ternary algebras. A ternary algebra is a de Morgan algebra with a third constant φ satisfying φ = φ-bar, and (a+a-bar)+φ=a-bar. If we define a third binary operation ∨ on T as a∨b=a*b+(a+b)*φ, then is a complemented semilattice.