State Pseudo-equality Algebras
Lavinia Corina Ciungu
Pseudo-equality algebras were initially introduced by Jenei and Kóródi as a possible algebraic semantic for fuzzy type theory, and they have been revised by Dvurečenskij and Zahiri under the name of JK-algebras. The aim of this paper is to investigate the internal states and the state-morphisms on pseudo-equality algebras.We define and study new classes of pseudo-equality algebras, such as commutative, symmetric, pointed and compatible pseudo-equality algebras. We prove that any internal state (state-morphism) on a pseudo-equality algebra is also an internal state (state-morphism) on its corresponding pseudo-BCK(pC) meet-semilattice, and we prove the converse for the case of linearly ordered symmetric pseudo-equality algebras. We also show that any internal state (state-morphism) on a pseudo-BCK(pC) meet-semilattice is also an internal state (state-morphism) on its corresponding pseudoequality algebra. The notion of a Bosbach state on a pointed pseudoequality algebra is introduced and it is proved that any Bosbach state on a pointed pseudo-equality algebra is also a Bosbach state on its corresponding pointed pseudo-BCK(pC) meet-semilattice. For the case of an invariant pointed pseudo-equality algebra, we show that the Bosbach states on the two structures coincide.
Keywords: Pseudo-equality algebra, pseudo-BCK algebra, pseudo-BCK meet-semilattice, pointed pseudo-equality algebra, compatible pseudo-equality algebra, symmetric pseudo-equality algebra, internal state, state-morphism, Bosbach state
AMS classification (2000): 03G25, 06F05, 06F35