A Characterization of Finite Ternary Algebras

A ternary algebra is a bounded distributive lattice with additonal operations e and ∼ that satisfies (a + b) ∼ = a ∼ b ∼ , a ∼∼ = a, e ≤ a + a ∼ , e ∼ = e and 0 ∼ = 1. This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternary algebra L, so that X freely generates L. With this characterization, the free ternary algebra on one free generator is displayed. The poset of join irreducibles of finitely generated free ternary algebras is characterized. The uniqueness of the set of free generators and their pseudocomplements is also established.

“…On the theoretical side, representation theorems and other basic properties have been developed by Brzozowski, J. J. Lou and Negulescu [3] and others cited in the bibliography.…”

Section: Introductionmentioning

confidence: 99%

Free Ternary Algebras

Balbes

2000

“…A representation theorem for finite ternary algebras, similar to the Stone representation theorem of Boolean algebras, has been proved in [4]. In this note, we drop the finiteness condition and derive the same result from the well-known representation of distributive lattices by lattices of sets.…”

Section: Introductionmentioning

confidence: 66%

“…We start by generalizing a construction from [4]. Let L = (L, V, A, 0,1) denote a bounded distributive lattice, so that L op , the opposite (or dual) of L and the direct product L op x L are also bounded distributive lattices.…”

Section: Representation By Set Algebrasmentioning

confidence: 99%

See 1 more Smart Citation

A Cayley Theorem for Ternary Algebras

Ésik

1998

“…For a short history of ternary algebras see Brzozowski, Lou, and Negulescu [4]. Here we follow Brzozowski and Seger [5] in defining a ternary algebra as a de Morgan algebra with an additional constant φ satisfying φ =φ and (a +ā) + φ = a +ā.…”

Section: Ternary Algebrasmentioning

confidence: 99%

Involuted Semilattices and Uncertainty in Ternary Algebras

Brzozowski

2004

Self Cite

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.