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Congruences on ideal nil-extension of completely regular semigroups 被引量:1
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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 《Chinese Science Bulletin》 SCIE EI CAS 1998年第5期379-381,共3页
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 cong... 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? 展开更多
关键词 COMPLETELY regular SEMIGROUPS IDEAL nil-extension congruences.
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