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Free algebras in varieties of Stonean residuated lattices. (English) Zbl 1142.06003
The variety of Stonean residuated lattices consists of all bounded (integral, commutative) residuated lattices \(\langle A, *, \rightarrow, \vee, \wedge, \top, \perp \rangle\) that satisfy the identity \(\neg \, x \vee \neg \neg \,x = \top\), where \(\neg \, x := x \rightarrow \, \perp\). A key feature of a Stonean residuated lattice is that double negation is a retract onto its Boolean skeleton. This paper presents a weak Boolean product representation of the free algebras in varieties of Stonean residuated lattices. Given a variety \(V\) of Stonean residuated lattices and a generating set \(X\), let \(V^*\) be the variety of all residuated lattices that become members of \(V\) when a least element is appended. Then the \(V\)-free algebra over \(X\) is represented as a weak Boolean product over the Cantor space \(2^{|X|}\) of the family of \(V^*\)-free algebras over subsets of \(X\), with a least element appended.

MSC:
06F05 Ordered semigroups and monoids
08B20 Free algebras
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