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; Sα >, each Sa satisfies the conditions of Lemma 2.2.1. If S satisfies conditions:(?)α,β ∈ D, 0α0β=0αβ,0α*0β=0αβ, (2.2.1)Va,β∈D,0α + 0β＝0αα+β, (2.2.2)a relation σ on S is defined by: a ∈ Sα b ∈Sβaσb(?) a0αβ3+b0αβ+a0αβ=a0αβ,b0αβ+a0αβ+b0αβ=b0αβ.Then σ is a bi-semiring congruence on S, and (S/σ, +) is a semilattice. Especially if (Sα, +)((?)α∈ D) is commutative, and satisfies the condition (?)a, b ∈ Sαif a0α = b0α, then a = b. (2.2.3)Then S is a subdirect product of an ide1potent semiring S0= {0α|α∈D} and a semiring S/σ .Theorem 2.3.2 Let S =< D; Sα >, {σα}α∈D are a family of admissible congruences on S, a relation σ on S is defined by: a E Sa, b ∈ Sβaσb(?) (a+0αβ,b+0α+β)∈σα+β.Then σ is a bi-semiring congruence on S.Corollary 2.3.4 Let {σα}α∈D are a family of admissible congruences on S=< D; Sα >, and a is the congruence on S correctly induced by {σα}α∈D thenσα=σ|Sα(α∈D).Lemma 2.4.1 Let S ＝< D; Sα >, and (?)δ≥α, Sδ(?) Sα+0δ, then C is the sublattice of ΠLα.Lemma 2.4.2 Let σ∈L1, then {σ|Sα}α∈D are a family of admissible con-gruences on S =< D; Sα > and σ is the congruence correctly induced by {σα}α∈D.Theorem 2.4.3 Let S ＝< D; Sα >, and (?)δ> α, Sδ(?)Sα+0δ.Define a map φ: C→L1, Πα∈D σα(?)σ, where σ a is the congruence on S correctly induced by {σα}α∈D, then φ is an isomorphism from C.Theorem 2.4.4 Let {σα}α∈D are a family of admissible congruences on S andσα is the bi-semiring congruence on Sα((?)α ∈D) with Sα(Sα/σα,+) is commutative,then σ that correctly induced by{σα}α∈D is the bi-semiring congruence on S with(S/σ, ＋) is commutative; Conversely, Let Sα(α∈ D) is E-inverse bi-semiring and a is the bi-semiring congruence on S with (S/σ, +) is commutative and satisfies:(?) a∈Sα,b∈Sβ,(?)δ≥α+β,(a+0δ,b+0δ)∈(?)(a,b)∈σ,then σ correctly induced by {σα}α∈D and Π∈D(σ|Sα)∈C,σ|Sα: is the bi-semiring congruence on Sa with (Sα/(σ|Sα), +) is commutative.Theorem 2.5.2 Let S =< D; Sα >,σ is the corresponding distributive lattice congruence on S, σ is a congruence on S, (?)α∈ D, Let σα＝σ|Sα, and satisfies (?)a, b ∈Sα,(?)δ≥α,(a + Oδ) b + Oδ)∈σδ(?) (a, b) ∈ σα.Then S/σ = S is the distributive lattice of bi-semirings with additional zero ele-ments {Sα/σα＝Sα}α∈D if and only if σ(?)σ.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; Sα],{σα}α∈D are a family of admissible con-gruences on S, a relation σ on S is defined by: α ∈ Sa, b ∈ Sβασb(?)(a· 1αβ, b·1αβ) ∈σαβThen σ is the bi-semiring congruence on S.Theorem 3.1.4 Let {σα}α∈D are a family of admissible congruences on S =[D; Sα], and σ is the congruence on S correctly induced by {σα} α∈D, Thenσα = σ|Sα{α ∈ D).Theorem 3.2.2 Let S = [D; Sα] σ is the quasidistributive lattice congruence on S, σ is a congruence on S, (?)α ∈ D, let σα＝ σ|Sα and (?)a, b∈Sαa,(?)δ≤α,satisfies(a· 1δ,b·1δ)∈σδ(?) (a, b)∈σα.Then S/σ=S is the quasidietributive lattice of bi-semirings with multiplication identical elements {Sα/σα=Sα}α∈D if and only if σ(?)σ.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; Sα], and satisfies: (?)a∈Sα b ∈ Sβ, α,β∈D.a · 1αβ=b ·1αβ(?)(?)≤(?)δ≤αβ, a·1δ＝b·1δ.(1)a relation σ on S is defined by: a ∈ Sα, b∈Sβ,α,β∈D aσb(?)a·1αβ= b· 1αβ.Then σ is the bi-semiring congruence on S, and S is a subdirect product of dis-tributive lattice D and bi-semiring S/σ.Theorem 4.2.7 Let S = [D; Sα],{(Nα,Tα)}α∈D are a family of I-normal congruence pairs on S. Let N=Uα∈D Nα τ,τ={(e,f) ∈E·(S) × E·(S)|e ∈E·(Sα), f ∈ E·(Sβ), (e·1αβ, f·1αβ) ∈α,β}, then (N, τ) 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; Sα], (N, τ) is I-standard congruence pair on S, let Na = N ∩Sα, τa=τ|E·(Sα), then {(Na, Tα)}α∈D are a family of I-normal congruence pairs on S, and (N,τ) is the I-normal congruence pair on S correctly induced by {(Nα,Tα)}α∈D.