The Structure And Congruence Of Some Bi-semirings

In this dissertation, we mainly dicuss the structures and congruences of bi-semirings, we mainly discuss the structures and congruences of a distributive lat-tice of bi-semirings with additional zero elements and a quasidistributive lattice of bi-semirings with multiplication identical elements and a quasidistributive lattice of inverse bi-semirings with multiplication identical elements. The dissertation is divided into four chapters:In the first chapter, We study the bi-semirings whose (S, +) semigroup are semilattices, (S, *) semigroup are semilattices and (S, ·) semigroup are rectangular groups. In order to prove Green - H -relation H on (S, ·) semigroup of S to be a bi-semiring congruence,we construct two partial order relations on S. Moreover,we give equivalent statements that H is a bi-semiring congruence. The main results are given in follow:Theorem 1.21 Let S?Sl?ReG?Sl, then H is bi-sermirings congruence on S if and only if (a +b )0=a0 +b0, (a *b6)0=a0 *b0,(?)a,b?S.Theorem 1.2.2 Let S?Sl ?ReG?Sl, then H is bi-sermirings congruence on S if and only if ?+=?0,?*=?0.Theorem 1.2.3 Let S ? Sl?ReG? Sl. Then the following statements are equivalent:(1) H is bi-sermirings congruence;(2) ?+??0,?*??0;(3) (?)a, b?S , (a +b)0=a0+b0, (a*b)0?a0*b0.In the second chapter, we mainly discuss the structures and congruences of a distributive lattice of bi-semirings with additional zero elements. The main results are given in follow:Theorem 2.2.2 Let S =< D; S? >, each Sa satisfies the conditions of Lemma 2.2.1. If S satisfies conditions:(?)?,? ? D, 0?0?=0??,0?*0?=0??, (2.2.1)Va,??D,0? + 0??0??+?, (2.2.2)a relation ? on S is defined by: a ? S? b ?S?a?b(?) a0??3+b0??+a0??=a0??,b0??+a0??+b0??=b0??.Then ? is a bi-semiring congruence on S, and (S/?, +) is a semilattice. Especially if (S?, +)((?)?? D) is commutative, and satisfies the condition (?)a, b ? S?if a0? = b0?, then a = b. (2.2.3)Then S is a subdirect product of an ide1potent semiring S0= {0?|??D} and a semiring S/? .Theorem 2.3.2 Let S =< D; S? >, {??}??D are a family of admissible congruences on S, a relation ? on S is defined by: a E Sa, b ? S?a?b(?) (a+0??,b+0?+?)???+?.Then ? is a bi-semiring congruence on S.Corollary 2.3.4 Let {??}??D are a family of admissible congruences on S=< D; S? >, and a is the congruence on S correctly induced by {??}??D then??=?|S?(??D).Lemma 2.4.1 Let S ?< D; S? >, and (?)???, S?(?) S?+0?, then C is the sublattice of ?L?.Lemma 2.4.2 Let ??L1, then {?|S?}??D are a family of admissible con-gruences on S =< D; S? > and ? is the congruence correctly induced by {??}??D.Theorem 2.4.3 Let S ?< D; S? >, and (?)?> ?, S?(?)S?+0?.Define a map ?: C?L1, ???D ??(?)?, where ? a is the congruence on S correctly induced by {??}??D, then ? is an isomorphism from C.Theorem 2.4.4 Let {??}??D are a family of admissible congruences on S and?? is the bi-semiring congruence on S?((?)? ?D) with S?(S?/??,+) is commutative,then ? that correctly induced by{??}??D is the bi-semiring congruence on S with(S/?, ?) is commutative; Conversely, Let S?(?? D) is E-inverse bi-semiring and a is the bi-semiring congruence on S with (S/?, +) is commutative and satisfies:(?) a?S?,b?S?,(?)???+?,(a+0?,b+0?)?(?)(a,b)??,then ? correctly induced by {??}??D and ??D(?|S?)?C,?|S?: is the bi-semiring congruence on Sa with (S?/(?|S?), +) is commutative.Theorem 2.5.2 Let S =< D; S? >,? is the corresponding distributive lattice congruence on S, ? is a congruence on S, (?)?? D, Let ????|S?, and satisfies (?)a, b ?S?,(?)???,(a + O?) b + O?)???(?) (a, b) ? ??.Then S/? = S is the distributive lattice of bi-semirings with additional zero ele-ments {S?/???S?}??D if and only if ?(?)?.In the third chapter: we mainly discuss the structures and congruences of a quasidistributive lattice of bi-semirings with multiplication identical elements. The main result is given in follow:Theorem 3.1.2 Let S = [D; S?],{??}??D are a family of admissible con-gruences on S, a relation ? on S is defined by: ? ? Sa, b ? S???b(?)(a· 1??, b·1??) ????Then ? is the bi-semiring congruence on S.Theorem 3.1.4 Let {??}??D are a family of admissible congruences on S =[D; S?], and ? is the congruence on S correctly induced by {??} ??D, Then?? = ?|S?{? ? D).Theorem 3.2.2 Let S = [D; S?] ? is the quasidistributive lattice congruence on S, ? is a congruence on S, (?)? ? D, let ??? ?|S? and (?)a, b?S?a,(?)???,satisfies(a· 1?,b·1?)???(?) (a, b)???.Then S/?=S is the quasidietributive lattice of bi-semirings with multiplication identical elements {S?/??=S?}??D if and only if ?(?)?.In the fourth chapter, we mainly discuss the structures and congruences of a quasidistributive lattice of inverse bi-semirings with multiplication identical ele-ments. The main result is given in follow:Theorem 4.1.5 Let S = [D; S?], and satisfies: (?)a?S? b ? S?, ?,??D.a · 1??=b ·1??(?)(?)?(?)????, a·1??b·1?.(1)a relation ? on S is defined by: a ? S?, b?S?,?,??D a?b(?)a·1??= b· 1??.Then ? is the bi-semiring congruence on S, and S is a subdirect product of dis-tributive lattice D and bi-semiring S/?.Theorem 4.2.7 Let S = [D; S?],{(N?,T?)}??D are a family of I-normal congruence pairs on S. Let N=U??D N? ?,?={(e,f) ?E·(S) × E·(S)|e ?E·(S?), f ? E·(S?), (e·1??, f·1??) ??,?}, then (N, ?) is the congruence pair on S.Theorem 4.2.9 The congruence pair that structure in Theorem 4.2.7 is I-standard congruence pair on S.Theorem 4.2.10 Let S = [D; S?], (N, ?) is I-standard congruence pair on S, let Na = N ?S?, ?a=?|E·(S?), then {(Na, T?)}??D are a family of I-normal congruence pairs on S, and (N,?) is the I-normal congruence pair on S correctly induced by {(N?,T?)}??D.