On right simple and right 0-simple ordered groupoids-semigroups (Q2707040)

This paper continues to generalize semigroup-theoretical results to the theory of partially ordered (p.o.) semigroups or groupoids. Here the relation between (right, left) simplicity and (right, left) \(0\)-simplicity is dealt with. A p.o. semigroup \((S,\cdot,\leq)\) without zero is called (right, left) simple if \(S\) is the only (right, left) ideal of \(S\) (in the semigroup- and order-theoretical sense). A p.o. semigroup \((S,\cdot,\leq)\) with zero is called (right, left) \(0\)-simple if \(S^2\neq\{0\}\) and if the only (right, left) ideals of \(S\) are \(\{0\}\) and \(S\). It is shown that \((S,\cdot,\leq)\) without zero is (right, left) simple if and only if the p.o. semigroup \(S^0\) obtained by adjunction of a zero which is at the same time the least element of \(S^0\), is (right, left) \(0\)-simple. ``Conversely'' it is proved that a p.o. semigroup \((S,\cdot,\leq)\) with zero is (right, left) \(0\)-simple if and only if \(S\setminus\{0\}\) is a (right, left) simple subsemigroup of \(S\). The case of p.o. groupoids is also considered.