正则^＊—半群的覆盖

Covers for Regular  semigroups

设Ｓ是一个正则＊－半群，Ｃ＊（Ｓ）是Ｓ的最小自共轭全子半群．在Ｓ上定义关系ρ：ａρｂｕ，ｖ∈Ｃ＊（Ｓ）ｓ．ｔ．ｕ＊ｕ＝ａａ＊，ｕｕ＊＝ｂｂ＊，ｖ＊ｖ＝ｂ＊ｂ，ｖｖ＊＝ａ＊ａ，ｂ＝ｕａｖ．用Ｇ表示Ｓ／ρ的置换群，Ｐ（Ｇ）表示Ｇ非空子集的集合．τ是Ｓ到Ｐ（Ｇ）的映射满足条件：（１）ｓ１，ｓ２∈Ｓ，（ｓ１τ）（ｓ２τ）（ｓ１ｓ２）τ；（２）ｓ∈Ｓ，｛ｇ－１∈Ｇ：ｇ∈ｓτ｝ｓ＊τ；（３）１τ－１＝Ｃ＊（Ｓ）．则Ｔ＝｛（ｓ，ｇ）∈Ｓ×Ｇ：ｇ∈ｓτ｝是Ｓ的一个Ｃ＊－酉覆盖．称正则＊－半群Ｓ的一个子集Ｈ是允许的，如果关于任意ａ，ｂ∈Ｈ，ｕ，ｖ∈Ｃ＊（Ｓ），有ａ＊ｂ，ａｂ＊∈Ｃ＊（Ｓ）和ｕａ，ｂｖ∈Ｈ．用Ｃ（Ｓ）表示Ｓ的所有允许子集（注意到Ｃ（Ｓ）是逆半群）．设Ｓ是一个正则＊－半群，Ｇ是一个群．如果θ：ｇ→θｇ是Ｇ到Ｃ（Ｓ）的一个准同态满足∪ｇ∈Ｇθｇ＝Ｓ，则Ｔ＝｛（ｓ，ｇ）∈Ｓ×Ｇ：ｓ∈θｇ｝是Ｓ的一个Ｃ＊－酉覆盖且Ｔ／σＧ．反之，Ｓ的每一个Ｃ＊－酉覆盖都可以如此构造．

Let S be a regular  semigroup and C (S) be the least self conjugate full subsemigroup. Define a relation ρ on S byaρbu,v∈C (S) s.t. u u=aa ,uu =bb ,v v=b b,vv =a a,b=uav.Denote by G the permutation group of S/ρ and P(G) the set of all nonempty subsets of G. Let τ be a mapping from S to P(G) satisfying:(1)s 1,s 2∈S,(s 1τ)(s 2τ)(s 1s 2)τ;(2)s∈S,{g -1 ∈G:g∈sτ}s τ;(3)1τ -1 =C (S). Then T={(s,g)∈S×G:g∈sτ} is a C  unitary cover of S.A nonempty subset H of a regular  semigroup S is permissible if a b,ab ∈C (S) and ua,bv∈H for all a,b∈H,u,v∈C (S). Denote by C(S) all permissible subsets of S (note that C(S) is an inverse semigroup).Let S be a regular  semigroup and G a group. If θ:g→θ g is a prehomomorphism from G to C(S) such that ∪ g∈G θ g=S, then T={(s,g)∈S×G:s∈θ g} is a C  unitary cover of S and T/σG. Conversely, every C  unitary cover of S can be so constructed.