Multipliers on a nearlattice. (English) Zbl 0605.06005

By a nearlattice is meant is this paper a lower semilattice in which any two elements have a least upper bound whenever they have a common upper bound. In a nearlattice N with 0 \(a\nabla b\) is defined to mean that \(a\wedge b=0\) and \(((a\wedge x)\vee (x\wedge y))\wedge b=x\wedge y\wedge b\) for all x,y\(\in N\). For a subset H of N with 0 let \(H^{\nabla}=\{a\in N\); \(a\nabla b\) for all \(b\in H\}\). - N is said to be the direct sum of \(H_ 1,...,H_ n\) (\(\subseteq N)\) if every element a of N can be expressed in the form \(a=a_ 1\vee...\vee a_ n\) \((a_ i\in H_ i)\) and \(H_ i\subseteq H_ j^{\nabla}\) whenever \(i\neq j\); the subsets \(H_ 1,...,H_ n\) are called direct summands of N. It is shown that for every element a of N the expression in a direct sum is unique and the direct summands are standard ideals of N; moreover, an ideal of N is a central element in the ideal lattice of N if and only if it is a direct summand of N. - A mapping \(\phi\) of a nearlattice N into itself is called here a multiplier on N if \(\phi (x\wedge y)=\phi (x)\wedge y\) for each x,y\(\in N\). By means of multipliers there is given a condition which is necessary and sufficient for a nearlattice with 0 to have a direct decomposition.
Reviewer: G.Szász


06A12 Semilattices
06B10 Lattice ideals, congruence relations
06B05 Structure theory of lattices
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