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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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