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Paper IPM / M / 16529  


Abstract:  
In this article we introduce the notion of a Euclidean commutator
lattice L and study some of its algebraic properties in a
parallel fashion with Euclidean (rings, semirings, modules, and
semimodules). The notion of an almost division algorithm and join
absorptive subsets of order t ≥ 1 (t a fixed integer) in
L as a parallel extension of additively absorptive subsemirings
of Euclidean semirings are discussed. For a fixed integer t ≥ 1, a lower set D of a Euclidean commutator lattice L with
Euclidean function ϕ, is said to be join absorptive (or
simply, absorptive) of order t in L; if for each f in L\D, there exists h in D and g in L such that f = h ∨g with 1 ≤ ϕ(g) ≤ t. The main result of the
paper states that if I is an ideal of an absorptive subset D
of order t in L, then I can be generated by (t+1) or fewer
elements. In addition, if I contains an element of ϕ value
equals to i+ϕ(I) for some 1 ≤ i ≤ t with ϕ(I) = inf { ϕ(f)  f ∈ I}, then I cannot be a principal ideal in D
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