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INVOLUTED SEMILATTICES AND UNCERTAINTY IN TERNARY ALGEBRAS.
- Source :
-
International Journal of Algebra & Computation . Jun2004, Vol. 14 Issue 3, p295-310. 16p. - Publication Year :
- 2004
-
Abstract
- 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]
- Subjects :
- *LATTICE theory
*ALGEBRA
*TERNARY system
*MATHEMATICS
Subjects
Details
- Language :
- English
- ISSN :
- 02181967
- Volume :
- 14
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- International Journal of Algebra & Computation
- Publication Type :
- Academic Journal
- Accession number :
- 13506953
- Full Text :
- https://doi.org/10.1142/S0218196704001785