The Semidirect Products Congruences And Properties Of Some Semigroups

In chapter 1, we mainly discuss semidirect products of semigroups. They are depicted by idempotent methods in [3],[5]....We consider the important condition that the ralation J is a congruence in a completely regular semigroup. We can get better results than by idempotent methods. This can give us the suggestion that we should dig out the unique properties and characteristics when we research different types of semigroups. And we should take different methods of discussion to make the research in the domain richer. The main results are given in follow:Theorem 1.2.4 Let S. T be two semigroups,α:S→End(T), s (?)α(s) be a given homomorphism, the semidirect products S×_αT is a left regular orthogroup if and only if:(1)for every e∈E(S), t^e = t, where t∈T;(2)S and T are left regular orthogroups;(3)(?)t∈T, s∈S, then tJt^s.Theorem 1.2.6 Let S, T be two semigroups,α: S→End(T), s (?) a(s) be a given homomorphism, the semidirect products S×_αT is a left regular cryptogroup if and only if:(1)for every e∈E(S), t^e=t, where t∈T;(2)S and T are left regular orthogroups;(3)(?)t∈, s∈S, then tJt^s;Theorem 1.3.2 Let the semidirect products S×_α, Tis a left regular cryptogroup,σS×_αTis the least group congruence, then Theorem 1. 3. 3 Let semidirect products S×_αTis a left regular cryptogroup, for every s₁, s₂∈S, t₁, t₂∈T, then(1)(s₁,t₁)L^S×_aT(s₂, t₂) (?) S₁L^Ss₂, t₁L^Tt₂.(2)(S₁,t₁)J^Sx_αT(s₂, t₂) (?) S₁J^Ss₂, t₁J^Tt₂.In chapter 2, Guo Yuqi showed that every completely semiprime congruence is an intersection of completely prime congruences on an arbitrary semigroup. Shumk. p and XiangYun Y dicuss n-prime ideals on A-semilattics and commutative ordered semigroup differently. In this paper, prime and semiprime congruences are considered. It is showed every semiprime congruence is an intersection of prime congruences. The main results are given in follow:Theorem 2. 2. 6 Let S be a semigroup. Then the following statements are equivalent on S:(1)A Rees congruence on S is semiprime if and only if it is an intersection of a family of prime Rees congruences;(2)S is a weakly reduced semigroup if and only if S is isomorphic to the subdirect product of a family of weakly integral semigroups;(3)A congruenceρon a semigroup S is semiprime if and only ifρis an intersection of a family of prime congruences on S.Lemma 2. 3. 4 If F is a subsemigroup of S. Then F is a n-filter and for x∈S. xⁿ∈F implies x∈F if and only if S\F is a n-prime ideal of S.Theorem 2. 3. 10 Let S be a semigroup. Then the following statements are equivalent:(1) Every completely semiprime and n-prime Rees congruence is an intersection of a family of (n-1)-prime Rees congruences on S;(2)Every reduced and n-integral semigroup is a subdirect product of a family of (n-1)-integral semigroups;(3)Every completely semiprime and n-prime congruence is an intersection of a family of (n-1)-prime congruences on S.The relations betweenÏ€-regular semigroups and ideals are researched in chapter 3. We have studied ideals, two-sided principal ideals,Ï€-regular semigroups,Ï€-inverse semigroups,Ï€- orthogroups, stronglyÏ€-inverse semigroups and so on. We depictÏ€-inverse semigroups,Ï€- orthogroups, stronglyÏ€-inverse semigroups by two-sided principal ideals. The main results are given in follow:Theorem 3. 2. 2 Let S be a semigroup, Then(1) S is leftÏ€-inverse if and only if every two-sided principal ideal of S is leftÏ€-inverse;(2) S is rightÏ€-inverse if and only if every two-sided principal ideal of S is rightÏ€-inverse;(3) S isÏ€-inverse if and only if every two-sided principal ideal of S isÏ€-inverse.Theorem 3. 2. 3 Let S be a semigroup. Then(1) S isÏ€-orthodox if and only if RegS is a subsemigroup of S and every two-sided principal ideal of S isÏ€-orthodox;(2) S is stronglyÏ€-inverse if and only if RegS is a subsemigroup of S and every two-sided principal ideal of S is stronglyÏ€-inverse.