Varieties of commutative integral bounded residuated lattices admitting a Boolean retraction term
A residuated lattice is an algebra \(\mathbf{{A}}=(A;*,\rightarrow, \vee, \wedge, \bot,\top)\) of type \((2,2,2,2,0,0)\) such that \((A;*,\top)\) is a commutative monoid, \(\mathbf{L(A)}=(A;\vee,\wedge, \top)\) is a lattice with greatest element \(\top\), and the following residuation condition holds: \(x*y\leq z\) iff \( x\leq y\rightarrow z\), where \(x,y,z\in A\) and \(\leq\) is the natural order given by the lattice structure of \textbf{A}. The residuated lattices form a variety denoted by \(\mathbb{RL}\). The subvariety of bounded residuated lattices will be denoted by \(\mathbb{BRL}\). If \(\mathbb{V}\) is a variety of bounded residuated lattices, a Boolean retraction term for \(\mathbb{V}\) is a unary term \(t\) in the language of bounded residuated lattices such that for every \(\mathbf{A} \in \mathbb{V}\), \(t^{\mathbf{A}}\) (the interpretation of the term on \(\mathbf A\)) defines a retraction from \(\mathbf{A}\) onto \(\mathbf{B(A)}\) (the set of all Boolean elements of \(\mathbf{A}\)). The aim of this paper is to investigate subvarieties of \(\mathbb{BRL}\) admitting a Boolean retraction term and Theorem 2.4 presents a necessary and sufficient condition in this sense. In Section 3, using the notion of radical of a bounded residuated lattice \(\mathbf{A}\) (that is, the intersection of its maximal implicative filters), the authors define the radical algebra \(\mathbf{r}(\mathbf{A})\) of \(\mathbf{A}\) and, for a class \(\mathbb{K}\) of bounded residuated lattices, the radical class associated to \(\mathbb{K}\) as \(\mathbb{K}^r=\mathbb{I}\{\mathbf{r(A)}: \mathbf{A}\in \mathbb{K}\}\), i.e., the class of isomorphic copies of algebras in \(\{\mathbf{r(A)}: \mathbf{A}\in \mathbb{K}\}\). Theorems 3.8 and 3.9 from this section characterize \(\mathbb{V}^r\), where \(\mathbb{V}\) is a subvariety of \(\mathbb{BRL}\), in two different situations: firstly, if \(\mathbb{V}\) admits a Boolean retraction, then \(\mathbb{V}^r\) is closed under homomorphic images and direct products; secondly, if \(\mathbb{V}\) is involutive, admitting a Boolean retraction, then \(\mathbb{V}^r\) is a variety. In Section 4 the authors give an explicit description of the representation of a bounded residuated lattice as Boolean product of directly indecomposables and in Section 5 they present examples of varieties with a Boolean retraction term.
scientific article; zbMATH DE number 6136994