Strong semilattices of completely simple semigroups (Q1357253)

A construction of normal bands of groups \(S\) (i.e., strong semilattices \(Y\) of completely simple semigroups \(S_\alpha\), \(\alpha\in Y\)) is given. An essential ingredient in it is the natural partial order \(\leq_N\) on \(S\) which in this case is compatible with multiplication (on both sides) and has the property that the intersection of any principal order ideal of \((S,\leq_N)\) with an arbitrary component \(S_\alpha\) of \(S\) is either empty or consists of a single element. These properties are used to define on a disjoint union \(S\) of completely simple semigroups \(S_\alpha\) (\(\alpha\in Y\)) a multiplication \(*\), which makes \(S\) a normal band of groups. It is interesting to note that the arbitrary partial ordering \(\leq\), which is used in this construction and satisfies the above conditions, is then the natural partial order \(\leq_N\) on \((S,*)\). Also, it is shown that a completely regular semigroup \(S\) (i.e., a semilattice \(Y\) of completely simple semigroups \(S_\alpha\), \(\alpha\in Y\)) is a normal band of groups (i.e., is strong) if and only if for every \(a\in S_\alpha\) and every \(\beta\leq\alpha\) in \(Y\) the intersection of any principal order ideal \((a]\) of \((S,\leq_N)\) with \(S_\beta\subseteq S\) consists of a single element.