A notion of branching rank for semilattices with descending chain condition (Q1110547)

The author considers semilattices satisfying the descending chain condition. He defines the rank of such a semilattice and proves that the rank of a subsemilattice is at most that of the semilattice. It is shown that such a semilattice of finite rank n has at least \(2^ n-1\) elements. He defines the poset (respectively semilattice) dimension of such a semilattice and introduces the notion of (rank) \(\alpha\)- irreducible semilattices for an ordinal \(\alpha\) in terms of the irreducible elements. For every ordinal \(\alpha\), it is shown that there is a binary tree of rank \(\alpha\) and an \(\alpha\)-irreducible semilattice. He proves that every semilattice of finite rank n contains a finite subsemilattice which is n-irreducible. The level of a semilattice is then defined in terms of the irreducible elements, and it is proved that for a tree S, one of \(\{\) rank S, level \(S\}\) is finite if and only if the other one is. Moreover, in this case, if \(| S| >1\), then rank S\(=level S\).