INVOLUTED SEMILATTICES AND UNCERTAINTY IN TERNARY ALGEBRAS.

An involuted semilattice <S,∨,-> is a semilattice <S,∨> with an involution -: S→S, i.e., <S,∨,-> satisfies $\bar{\bar a}=a$, and $\overline{a\vee b}={\bar a}\vee{\bar b}$. 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 $\phi={\bar \phi}$, 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. [ABSTRACT FROM AUTHOR]