## 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

### MSC:

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