Simple characterization of strict residuated lattices with an involutive negation

Abstract

We consider property of strict residuated lattices (SRL-algebras) with a new involutive negation $\lnot,$ called here by SRL$_{\lnot }$-algebras, and give a simple characterization of SRL$_{\lnot }$-algebras. We also prove a prime filter theorem of SRL$_{\lnot }$-algebras, from which we conclude that every linearly ordered SRL$_{\lnot }$-algebra is simple. As a corollary to this fact, we have a well-known result that every SML$_{\lnot }$-algebra (SBL$_{\lnot }$-algebra) is a subdirect product of linearly ordered SML$_{\lnot }$-algebras (SBL$_{\lnot }$-algebras).