State-morphism algebras---General approach

Authors:
Michal Botur;Anatolij DvurečEnskij
Affiliations:
Department of Algebra and Geometry, Faculty of Science, Palacký University, 17. listopadu 12, CZ-771 46 Olomouc, Czech Republic;Department of Algebra and Geometry, Faculty of Science, Palacký University, 17. listopadu 12, CZ-771 46 Olomouc, Czech Republic and Mathematical Institute, Slovak Academy of Sciences, Št ...
Venue:
Fuzzy Sets and Systems
Year:
2013

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Abstract

We present a complete description of subdirectly irreducible state BL-algebras as well as of subdirectly irreducible state-morphism BL-algebras. In addition, we present a general theory of state-morphism algebras, that is, algebras of general type with state-morphism which is a fixed idempotent endomorphism. We define a diagonal state-morphism algebra and we show that every subdirectly irreducible state-morphism algebra can be embedded into a diagonal one. We describe generators of the varieties of state-morphism algebras, in particular generators of state-morphism BL-algebras, state-morphism MTL-algebras, state-morphism non-associative BL-algebras, and state-morphism pseudo MV-algebras.