Congruences on ideal nil-extension of completely regular semigroups

作     者：REN Xueming 1, 2 GUO Yuqi 2 and CEN Jiaping (Shum Kar-ping) 31. Department of Mathematics, Xi’an University of Architecture & Technology, Xi’an 710055, China 2. Department of Mathematics, Yunnan University, Kunming 650091, China 3. Department of 

作者机构：Department of Mathematics Xi’an University of Architecture & Technology Xi’an China Department of Mathematics Yunnan University Kunming China Department of Mathematics Chinese University of Hong Kong Hong Kong 

出 版 物：《Chinese Science Bulletin》 (科学通报(英文版))

年 卷 期：1998年第43卷第5期

页      面：379-381页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：ThisworkwassupportedbytheNationalNaturalScienceFoundationofChina (GrantNo .194710 6 8)  andbytheAppliedFoundamentalResearchFundationofYunnan (GrantNo .96a0 0 1z)andby 2 0 0 .6 0 0 .380oftheChineseUniversityofHongKong 

主　　题：completely regular semigroups ideal nil-extension congruences. 

摘      要：Let S be an ideal nil-extension of a completely regular semigroup K by a nil semigroup Q with zero. A concept of admissible congruence pairs (δ,ω) of S is introduced, where δ and ω are a congruence on Q and a congruence on K respectively. It is proved that every congruence on S can be uniquely respresented by an admissible congruence pair (δ,ω) of S. Suppose that ρ K denotes the Rees congruence induced by the ideal K of S. Then it is shown that for any congruence σ on S,a mapping Γ:σ|→(σ Q,σ K) is an order-preserving bijection from the set of all congruences on S onto the set of all admissible congruence pairs of S,where σ K is the restriction of σ to K and σ Q=(σ∨ρ K)/ρ K. Moreover,the lattice of congruences of S is also discussed. As a special case,every congruence on completely Archimedean semigroups S is described by an admissible quarterple of S. The following question is asked: Is the lattice of congruences of the completely Archimedean semigroup a semimodular lattice?