Some Quasi Completely Regular Semiring Structure And Congruence

This paper mainly deals with structures of some quasi-completely regular semirings and congruence on them. In ?,first offers a structure theorem of completely regular semigroups given by Mario Petrich(1987);secondly,offers structure of semiring ? ) while Et~1 E1~ and E satisfying some conditions, we mainly have the following results: Theorem If E1~1 = ~ in a semiring (S,+, . ),then S is a completely regular semiring if and only if S is a +梥emilattice semiring of completely simple semigroups. Theorem If (S,+, ) is a semiring and E~~1 ?E拁 then is a completely regular semiring and E is a band semiring if and only if S=ExDG= ?u(EaxGa),in aeD which E is a distributive lattice D of rectangular band semiring Ea and C is a distributive lattice D of group semiring Ga in ~3,offers structures of some quasi-completely regular semirings correspondent to the quasi-completely semigroups whose structures are described by Stojan Bodanovic in [4], we mainly have the following results: Theorem If ~ = E1~ in a quasi-completely inverse semiring (S,+, . ) and for any e, f e E the following conditions hold e + ef + e = e, ef + fe + ef = ef. then S is a distributive lattice of quasi-group semirings. Theorem If a semiring (S,+, . ) satisfies the condition E1~1 = ~ ,then S is a quasi-completely inverse semiring if and only if S is a +梥emilattice seiniring of quasi-group semiring. Theorem If (S,+, ? is a seniiring and ~ = ~ in it, then S is a quasi- completely regular semiring and E is a band semiring if and only if S is a distributive lattice of completely Archimedean semiring and Et~~ is a subsemigroup. Theorem If E1~1 = ~ in a semiring (S,+, . ), then S is a completely Archimedean semiring if and only if S is a nil 梕xtension of completely simple semiring. Theorem If a semiring (S,+, . ) have the property E拁~ ?E11, then S is a left 28 Clifford quasi-rectangular semiring if and only if S is a quasi-completely regular semiring and Re gS is a left Clifford semiring. In ?,according to the description of the minimum group congruence on some semigroups by Howie in [2] and on some quasi-completely regular semigroups by Stojan Bodanovic in [4],offers the minimum ring congruence on some quasi- completely semirings, the following are the main results: Theorem if we define the relation p in a additively inverse semiring (S,+, ) as follows: p = {(a,b) ~ Sx S~ (~e ~ E)a + e = b + e}. then p is the minimum ring congruence on S. Theorem If the relation p in additively quasi-inverse semiring (S,+, ) are defined as follows: p={(a,b)eSxS (~e~ E)a眅 =b+e}. then p is the minimum ring congruence on S.