Abstract:

\bg{abstract} By a symmetric residuated lattice we understand an
algebra $\A = (A, \lor, \land, \ast, \to, \sim, 1, 0)$ such that
$(A, \lor, \land, \ast, \to, 1, 0)$ is a commutative integral
bounded residuated lattice and the equations $\sim \sim x = x$ and
$\sim (x \lor y) = \sim x \land \sim y$ are satisfied. The aim of
the paper is to investigate properties of the unary operation
$\varepsilon$ defined by the prescription $\varepsilon x := \sim x
\to 0$. We give necessary and sufficient conditions for
$\varepsilon$ being an interior operator. Since these conditions are
rather restrictive (for instance, on a symmetric Heyting algebra
$\varepsilon$ is an interior operator if and only  the equation $(x
\to 0) \lor ((x \to 0) \to 0) = 1$ is satisfied) we consider when an
iteration of $\varepsilon$ is an interior operator. In particular we
consider the chain of varieties of symmetric residuated lattices
such that the $n$ iteration of $\varepsilon$ is a boolean interior
operator. For instance, we show that these varieties are semisimple.
When $n = 1$, we obtain the variety of symmetric stonean residuated
lattices. We also characterize the subvarieties  admitting
representations as subdirect products of chains. These results
generalize and in many cases also simplify, results existing in the
literature.\e{abstract}

• Tagged
• XML
• BibTex