Free Algebras Over a Poset in Varieties

Abstract

In 1945, the notion of free lattice over a poset was introduced by R. Dilworth (Trans. Am. Math. Soc. 57 (1945), 123-154). In this note, a construction of the free algebra over a poset in varieties finitely generated is shown. Finally, this result is applied to different classes of algebras.

Notes on the Variety of Ternary Algebras

“…A. Figallo Jr. and A. Ziliani in [8] construct the free algebra over a poset in finitely generated varieties; by using a similar argument, we indicate a construction of the free ternary algebra over a poset.…”

Free Algebras Over a Poset in Varieties of Łukasiewicz–Moisil Algebras

“…Later, this notion was adapted to different classes of algebras that arise from non-classical logics, these classes constitute varieties of algebras, which have an underlying order structure definable by means of certain equations p i px, yq " q i px, yq, 1 ≤ i ≤ n, in terms of the algebra's operations and some positive integer n. Constructions of this particular free algebra have been exhibited for different kinds of algebras such as bounded distributive lattices, De Morgan algebras and Hilbert algebras (see [7,8]). …”

“…A general construction of the free algebra over a poset in varieties finitely generated is given in [8]. In this paper, we apply this to the varieties of Łukasiewicz-Moisil algebras, giving a detailed description of the free algebra over a finite poset pX, ≤q, Free n ppX, ≤qq.…”