Luiz F. Monteiro

Notes on Free Monadic Boolean Algebras

If F B(2n − 1) denotes the Boolean algebra with 2n − 1 free generators and P(2n) is the Cartesian product of 2n Boolean algebras all equal to F B(2n − 1), we define on P(2n) an existential quantifier ∃ by means of a relatively complete Boolean subalgebra of P(2n) and we prove that (P(2n),∃) is the monadic Boolean algebra with n free generators. Every element of P(2n) is a 2n-tuple whose coordinates are in F B(2n − 1); in particular, so are the n generators of P(2n). We indicate in this work the coordinates of the n generators of P(2n).

Construction of monadic three-valued Łukasiewicz algebras

AbstractThe notion of monadic three-valued Łukasiewicz algebras was introduced by L. Monteiro ([12], [14]) as a generalization of monadic Boolean algebras. A. Monteiro ([9], [10]) and later L. Monteiro and L. Gonzalez Coppola [17] obtained a method for the construction of a three-valued Łukasiewicz algebra from a monadic Boolea algebra. In this note we give the construction of a monadic three-valued Łukasiewicz algebra from a Boolean algebra B where we have defined two quantification operations ∃ and ∃* such that ∃∀*x=∀*∃x (where ∀*x=-∃*-x). In this case we shall say that ∃ and ∃* commutes. If B is finite and ∃ is an existential quantifier over B, we shall show how to obtain all the existential quantifiers ∃* which commute with ∃.Taking into account R. Mayet [3] we also construct a monadic three-valued Łukasiewicz algebra from a monadic Boolean algebra B and a monadic ideal I of B. The most essential results of the present paper will be submitted to the XXXIX Annual Meeting of the Unión Matemática Argentina (October 1989, Rosario, Argentina).

Free Monadic Tarski and MMI3-Algebras

Abstract MMI3-algebras are a generalization of the monadic Tarski algebras as defined by A. Monteiro and L. Iturrioz, and a particular case of the MMIn+1-algebras defined by A. Figallo. They can also be seen as monadic three-valued Łukasiewicz algebras without a first element. By using this point of view, and the free monadic extensions, we construct the free MMI3-algebras on a finite number of generators, and indicate the coordinates of the generators. As a byproduct, we also obtain a construction of the free monadic Tarski algebras.

Finite free generating sets

Abstract In this paper we find the number of free generating sets with q elements for free algebras in the varieties of Boolean algebras, three-valued Łukasiewicz algebras, (0, 1)-distributive lattices and De Morgan algebras. For the varieties of Boolean algebras, three-valued Łukasiewicz algebras and Post algebras we also provide an explicit construction of a finite free generating set.

Free three-valued Lukasiewicz, Post and Moisil algebras over a poset

A construction of the free three-valued Lukasiewicz algebra L(L) over a poset L is given. The number of elements of L(L) for some particular cases of finite posets L is determined. The free (three-valued) Post and Moisil algebras over a poset are determined.<<ETX>>