On Structures Of Semirings And Congruences On Semirings

In this dissertation, we give a definition of congruence pair of inverse semiring;besides ,we give a extent of congruence pair of inverse semiring;finally,we give do a series of generalising. The main results are given in follow.In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we give a definition of congruence pair of inverse semiring at first;besides we discuss a extension of congruence pair of inverse semiring.The main results are given in follow.Define 2.2 Let ρ be a semiring congruence of inverse semiring,we define the kernel and trace of ρ byrespectively.Define 2.4 Let R be an inverse semirong .A subset K of S is full if E(S) (?) K;it is self — conjugate if —s + K + s (?) K for all s ∈ S . A full ,self-conjugate inverse subsemiring of S is a normal subsemiring of R. A semiring conrguence r on E(R) is normal if for any e, f ∈ E(R) and r ∈ R, eτf implies (—r + e + r)τ(—r + f + r).The pair (K, τ) is a congruencepair for R if K is anormal subsemiring of R, r is a normal congruence on E(R).Theorem 2.6 Let R be an inverse semiring.If (K, τ) is a semiring congruence pair for R,then ρ(κ,τ) is the unique semiring congruence ρ on R for which ker ρ = K and tr ρ = τ. Conversely,if p is a semiring congruence on R,then (ker ρ, tr ρ) is a congruence pairfor R and p(fcerPi trp) = p.Theorem 2.12 Let S be an inverse semiring.And Ve G E(S),c G S,3f,g G E(S),we get e = cf = gc,Define a mapping tr bytr : p —>■ ￡rp [p GThen tr is a complete homomorphism of C(S) onto K(E(S)). Then for any p 6 €In chapter 3,we give a definition of special semiring congruence of an inverse semiring at first and we investigate the properties of them;finally,wo obtain a series of conclusions. The main results are given in follow.Theorem 3.1 Let R be an additively commutative inverse semiring,a, b G R, definiting o : (a, b) G o <^=^ a + e = b + e,3eG E(R), then a is a ring congruence ofR.Theorem 3.4 Let (R,+,.) be a semirin.Suppose that (R,+) is a group (not commutive), R is a subgroup of (R, -f),then R is a ideal of R.Theorem 3.5 Let R be an inverse semiring.Suppose that a is a ring congruence of R like as definition 3.1,then Keru = {a G R | aa = (a + a)a} is a full dense reflexive unitary ideal of R.Theorem 3.6 Let R be an inverse semiring.Suppose that (a, 6) G a <~> 3e G E(R), a + e — b + e,and Rl — R/a;(R[, +)is the additional commutator subgroup of (Ri, +), then R = Ri/R\ is the maxmal ring homomorphism image of R, and9 : (a,b) e9 <=> aa - ba G J?iis the minimal ring congruence.In chapter 4,we give the conditions of an inverse semiring how to expressed a subdirect prouduct of a ring and an additively idempotent semiring at first , and we investigate the properties of them;finally,wo obtain a series of conclusions. The main results are given in follow.Theorem 4.1 An inverse semiring 5 is a subdirect prouduct of a ring and an additively semiring if and only if E(5) is a K-ideal of S.Theorem 4.2 Let S be an inverse semiring.lt can be said a subdirect prouduct of a ring R and an additively idempotent semiring T. Vr G R, b € T,then rb.br € T.we can get any congruence a like as :(a, b)a(c, d) <(=? a — c G IAR, bpd,p is a semering congturnce of T.