Deductive Systems and Congruences in RM Algebras
The class of all RM algebras contains BCK, BCI, BCH, BZ, RME, pre- BZ algebras and many others. In this paper, we introduce the notion of translation deductive system in RM algebras and investigate its elementary properties. We prove that every closed translation deductive system is a congruence kernel. Moreover, we give the construction of quotient algebra 𝒜/D of a RM algebra 𝒜 via a translation deductive system D of 𝒜. Finally, we consider commutative RM, RME and pre-BZ algebras and show that commutative RME algebras coincide with commutative pre-BZ algebras and with commutative BCK algebras. We also show that there is a bijection between congruences and closed translation deductive systems in commutative RM algebras.
Keywords: RM/RME/BCI/BCH/BZ/pre-BZ/pre-BBBZ algebra, (translation) deductive system, congruence
Mathematics Subject Classification (2010): 03G25, 06F35, 06A06