正格序半群的格序和

ON LATTICE-ORDERED SUM OF POSITIVE LATTICE-ORDERED SEMIGROUPS

本文将〔1～2〕中关于全序半群的有序和的讨论推广到格序半群的情况,由一类正格序半群构造出一新的正格序半群并讨论了正格序半群的一些性质。

Let Γ be a upper semilattice. To each λ∈Γ we assign a positive lattice-orderedsemigroup with the least element e_λ such that for λ≠μ,S_λ and S_μ are disjoint, we denoted it byS=U λ∈Γ S_λ. We define an opration '。' in S as follows: If a∈S_λ, b∈S_μ, and λ=μ, then the opera-tion in S is the same as in S_λ, otherwise a?b=b?a={a if λ>μ b if λ<μ e_γ if λ‖μ and γ=λ?μThen we have Theorem I. S=Uλ∈Γ S_λ about multiplication '。' is a semigroup. We call S as lattice-orderedsum of {s_λ}_(λ∈Γ). It is an extension of orderial sum about total order semigroups in [1, 2]. Theorem 2. Define an ordering relation '≤' in S as follows: if λ=μ,the ordering relatio-nis the same as in S_λ; if λ<μ, then put a≤b. then S is a lattice-ordered semigroup about '≤' in other parts,we give some properties of lattice-ordered sum.