Semirings and pseudo MV algebras

Abstract

In this paper, we describe the relationships between pseudo MV algebras and semirings. We also give definitions of automata on lattice ordered semirings, prove that the family of K-Languages is closed under union, and discuss the conditions for the closedness of families of K-languages under intersection, generalized intersection and reversal operations.

Automata theory based on lattice-ordered semirings

“…Obviously if the alphabet of the K automata is R, rather than R , the K automata defined above is exactly the K À R automata in (Shang and Lu 2007).…”

“…Note that the empty-move was not considered in (Shang and Lu 2007). The K À R automaton defined in Shang and Lu (2007) is the KNFA in this paper except that the value of accepted by automaton is defined here, while its value is missed in Shang and Lu (2007).…”

“…In this paper, we discuss more properties of automata and languages based on lattice-ordered semirings besides the union, intersection and reversal of languages in (Shang and Lu 2007). In detail, we give the definitions of KDFA, KNFA, K regular expressions and K regular grammars.…”

“…Further, Shang and Lu (2007) discussed the semiring reducts on pseudo MV algebras, established the relation between pseudo MV algebras and lattice-ordered noncommutative semirings. They set up automata theory based on these non-commutative algebraic structures.…”

“…Definition 2.5 (Shang and Lu 2007) A right _-coupled semiring A is a structure ðR 1 ; R 2 ; a; bÞ such that (i) R 1 ¼ ðA; _; 0; Á; eÞ and R 2 ¼ ðA;^; e; Á 0 ; 0Þ are a lattice-ordered semiring and a dual lattice-ordered semiring respectively,…”