Selfdistributive groupoids. AI: Non-idempotent left distributive groupoids. (English) Zbl 1080.20060

The authors describe the essentials (known results) of the algebraic theory of (generally non-idempotent) left distributive groupoids. A groupoid \(G\) is said to be left distributive (LD) if it satisfies the identity \(x\cdot yz=xy\cdot xz\). The theory of non-idempotent LD-groupoids has its own flavour and some of these groupoids are of special interest.
Chapter I deals with groupoids (stable relations and congruences, ideals, closed subgroupoids, regular groupoids, some varieties of groupoids. The right form of the identity of right alternativity is: \(yx\cdot x=y\cdot xx\).) Chapter II is concerned with the basic properties of LD-groupoids, with their ideals, dense subgroupoids, cancellable and divisible elements, with simple LD-groupoids. Chapter III is devoted to subdirect decompositions of some non-idempotent LD-groupoids. In Chapter IV various constructions and examples of LD-groupoids are described, and finally (up to isomorphisms) any two- and three-element LD-groupoid is given with its Cayley-table. One can find at the end of each chapter some comments and open problems, too.


20N02 Sets with a single binary operation (groupoids)
20N05 Loops, quasigroups
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