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Lattice ordered binary systems. (English) Zbl 0667.06008

This paper deals with a generalization of lattice (ordered) loops, namely with divisibility semiloops, which are defined to be semilattice-ordered cancellation groupoids with unit fulfilling ax\(\leq b\Rightarrow \exists u: au=b\), and ya\(\leq b\Rightarrow \exists v: va=b\). As one main motivation for studying these structures, the author shows that the lattice loop cones (i.e. the sets of those p with \(x\leq px,xp\) for all x) are exactly the positive divisibility semiloops, and that they can be realized as cones of loops fulfilling the inverse property.
Generalizing a result of T. Evans, he determines all divisibility semiloops satisfying the descending chain condition for intervals \([a,b)\) to be the direct sums of copies of \(({\mathbb{Z}},+,\min)\) and \(({\mathbb{N}}^ 0,+,\min).\) Giving an analogue of K. Iwasawa’s theorem, he shows that any power-associative (but not any non-associative) complete divisibility semiloop is associative and commutative. Further, he characterizes those divisibility semiloops which admit a complete extension.
The main part of the paper is devoted to decomposition theorems in the spirit of Stone’s classical result on representable Boolean algebras. The author proves that a divisibility semiloop is representable (i.e. a subdirect product of totally ordered factors) if and only if \(p(a)\wedge q(b)\leq p(b)\vee q(a)\) for any pair \(p,q\) of multiplication polynomials. Moreover, he extends this result to arbitrary lattice ordered algebras \((A,\wedge,\vee,f_ i)\) where each operation is isotone (and has to distribute over meet and join) at each place. This solves some problems stated by L. Fuchs (1966) and T. Evans and P. A. Hartman (1983), and yields a wide variety of specializations and corollaries, concerning lattice semigroups and loops, Holland’s theorem on lattice groups, distributive lattice monoids, complementary semigroups, dually residuated semigroups, lattice rings, and cone algebras.
Reviewer: F.Kalhoff

MSC:

06F05 Ordered semigroups and monoids
20N05 Loops, quasigroups
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
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