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Right ideals in a right distributive groupoid. (English) Zbl 0544.20052
Let G denote a groupoid, $$<G>$$ denotes the groupoid generated by G under set product [e.g. $$GG^ 2\in<G>$$ where $$GG^ 2=\{a(bc)| \quad a,b,c\in G\}],$$ R(G) denotes the set of right ideals of G, and $$P(G)=\{G^ n|$$ n is a positive integer} where $$x^{n+1}=x^ nx$$ for $$x\in G$$. It is well known that if G is a semigroup then: (i) R(G) is a semigroup under set product, (ii) $$<G>\subseteq R(G)$$, (iii) $$<G>$$ is totally ordered under inclusion. In general (i), (ii), (iii) are not true. However, in this paper, it is shown that if G is a right distributive groupoid [i.e. $$(xy)z=(xz)(yz)]$$ then R(G) is a right distributive groupoid and conditions (ii) and (iii) are satisfied. Examples indicate that, in general, $$P(G)\neq<G>$$ and that $$<G>$$ is right distributive does not imply G is right distributive (although the converse is true). The following are equivalent: (a) $$<G>$$ is right distributive; (b) if $$Y,V\in<G>$$ such that $$Y\neq G$$ then $$YV=YG$$ and $$(GV)G=G^ 3$$; if $$A,B,C\in<G>$$, then $$(AB)C=(AG)G$$. Finally if $$<G>$$ is right distributive the following conditions, on $$<G>$$, are characterized: $$<G>=P(G)$$, commutativity, associativity, distributivity (both sides).
##### MSC:
 20M10 General structure theory for semigroups 20M12 Ideal theory for semigroups
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##### References:
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