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Involutive Monoidal T-norm Based Algebras with Internal States
Pengfei He, Juntao Wang and Bin Zhao

In the paper, we investigate internal states on involutive monoidal tnorm based algebras, which are algebraic semantics of the logic of left-continuous t-norms with an involutive negation and their residua. The resulting class of algebras will be called IMTL-algebras with internal states or state IMTL-algebras. First, we discuss some properties of internal states and obtain relationships between states and internal states on IMTL-algebras. Moreover, we prove that if an IMTL-algebra L is termwise equivalent to an MV-algebra, then an internal state on L taken in the IMTL-setup coincides with the notion of an internal state in the MV-setup. Secondly, we introduce and characterize maximal state filters and prime state filters in state IMTL-algebras. Using them, we prove that subdirect representation theorems of pre-simple state IMTLalgebras and state pre-linear state IMTL-algebras. Furthermore, we characterize subdirectly irreducible state IMTL-algebras, which is faithful. Finally, studying some topological structures on the set Specτ [L] of all prime state filters and the set Maxτ [L] of all maximal state filters in a state IMTL-algebra, respectively, we conclude that Specτ [L] is a compact T0 topological space and Maxτ [L] is a compact Hausdorff topological space. In particular, we introduce pure state filters in state IMTL-algebras and characterize stable open sets in Specτ [L]. Also, we obtain that Specτ [L] with the stable topology is a compact topological space but not T0.

Keywords: IMTL-algebra; internal state; state filter; pure state filter; stable open set; spectral/stable topology

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