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On algebra-lattices. (English) Zbl 0579.08002

In this article an algebra-lattice is an ordered universal algebra (A,F,\(\leq)\) which is a complete lattice and satisfies f(a,...,a)\(\leq a\) and \(f(1,...,1,0,1,...,1)=0\) for any \(f\in F\) and any \(a\in A\). An element \(b\in A\) is called an f-absorbent (f-quasiabsorbent) of \(a\in A\) iff \(b\leq a\) and f(a,...,a,b,a,...,a)\(\leq b\) for all \(i=1,...,n\); here b is in the ith place \((\wedge^{n}_{i=1}f(a,...,a,b,a,...,a)\leq b)\); b is an absorbent (quasiabsorbent) of a iff it is an f-absorbent (f- quasiabsorbent) of a for any \(f\in F\). Some simple properties of absorbents and quasiabsorbents are derived, such as the following one: If \(b_ i\) (i\(\in I)\) are absorbents (quasiabsorbents) of a, then \(\bigwedge_{i\in I}b_ i\) is an absorbent (quasiabsorbent) of a.
Reviewer: V.Novák

MSC:

08A05 Structure theory of algebraic structures
06F25 Ordered rings, algebras, modules
06B23 Complete lattices, completions
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References:

[1] O. Steinfeld: Über Gruppoid-Verbände I. Acta Sci. Math., 31 (1970), 203-218. · Zbl 0212.36003
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