Ideals of semisimple MV-algebras and convergence along set-theoretic filters

Abstract

In this paper, we investigate the connection between ideals of semisimple MV-algebras and set-theoretic filters. In particular, we obtain that there is a bijection between closed ideals (with respect to a specified closure operation) and all filters on some associated set X. Finally, we establish that the space of closed prime ideals is homeomorphic to a well-described space having the Stone-Čech compactification βX as a subspace.

Introduction

MV-algebras are the algebraic counterparts of Łukasiewicz many-value logic. They can be defined [2] as algebras $〈A,\oplus ,⁎,0〉$ of type $(2,1,0)$ satisfying: (i) $〈A,\oplus ,0〉$ is an abelian monoid, (ii) $0^{⁎}\oplus x=0^{⁎}$, (iii) $x^{⁎⁎}=x$ and (iv) ${(x^{⁎}\oplus y)}^{⁎}\oplus y={(y^{⁎}\oplus x)}^{⁎}\oplus x$ for all $x,y\in A$. Each MV-algebra carries a natural distributive lattice order ≤ defined by $x\le y$ if and only if $x^{⁎}\oplus y=1$. An ideal of the MV-algebra A is a nonempty subset I of A satisfying (i) $x\oplus y\in I$ for all $x,y\in I$ and (ii) whenever $x\le y$ with $y\in I$, it follows that $x\in I$. A nontrivial MV-algebra A is said to be semisimple if the intersection of all its maximal ideals also known as its radical equals to {0}.

As in many algebraic theories and most notably in the commutative algebra, ideals play a key role in the study of the algebraic structure. MV-algebras are no exception to this approach. Indeed, many papers on MV-algebras have been devoted to the study of ideals and their different types. One cannot hope to achieve any significant results on ideals of general MV-algebras given the complexity of such algebras and their ideals. It has been customary to focus the studies either on special types of ideals such as prime ideals (see e.g., [1], [3], [5]) or on special classes of MV-algebras such as free MV-algebras (see e.g., [5, Section 3.4]). In this paper, we focus on the class of semisimple MV-algebras and wish to offer a description of the ideals based on set-theoretic filters. More precisely, given that semisimple MV-algebras are subalgebras of the bold fuzzy algebra ${[0,1]}^X$ for some nonempty sets X, we shall establish a connection between the ideals of such MV-algebras and the filters on X. The correspondence in question will be shown to behave nicely when one restricts the attention to the class of closed ideals, a class to be defined. This class of ideals is motivated by similar considerations in commutative algebra based on a fixed closure operation on the MV-algebra. Indeed, closure operations on MV-algebras were recently introduced [4] and can be used to study ideals of MV-algebras. In the present case, we consider a specific closure operation and investigate the closed ideals under the operation. We prove that these ideals are in one-to-one correspondence with the filters on X and that their overlap with prime ideals yields a space that contains a homeomorphic copy of the Stone-Čech compactification of X equipped with the discrete topology. Pushing the correspondence further down, we obtain a homeomorphism between the maximal spectrum of an MV-algebra and the Stone-Čech compactification of X equipped with the discrete topology. The paper is organized as follows. In Section 2, we start by reviewing the notion of limits along filters and use it to construct a correspondence between the ideals of the MV-algebras and set-theoretic filters. We establish a correspondence between the ideals of an MV-algebra $A\subseteq {[0,1]}^X$ and the filters on X. In Section 3, which is the main section of this paper, we introduce the class of closed ideals in a semisimple MV-algebra. We prove that maximal ideals of semisimple MV-algebras are closed. We construct a bijection between closed ideals of semisimple MV-algebras and the set of filters of a topological space X. We also exhibit a concrete homeomorphism between the maximal spectral $\mathrm{\text{Max}}(A)$ of a semisimple MV-algebras A and the Stone-Čech compactification βX of a discrete topological X. In addition, some properties of closed ideals of semisimple MV-algebras as well as some examples of ideals that are not closed are also treated. In Section 4, we have the closing remarks and future research.