We construct a countable chain of Boolean semilattices, with all inclusion maps preserving the join and the bounds, whose union cannot be represented as the maximal semilattice quotient of the positive cone of any dimension group. We also construct a similar example with a countable chain of strongly distributive bounded semilattices. This solves a problem of F. Wehrung.
The concept of covering energy of a poset is known and its McClelland type bounds are available in the literature. In this paper, we establish formulas for the covering energy of a crown with elements and a fence with elements. A lower bound for the largest eigenvalue of a poset is established. Using this lower bound, we improve the McClelland type bounds for the covering energy for some special classes of posets.
It is well known that the linear extension majority (LEM) relation of a poset of size can contain cycles. In this paper we are interested in obtaining minimum cutting levels such that the crisp relation obtained from the mutual rank probability relation by setting to its elements smaller than or equal to , and to its other elements, is free from cycles of length . In a first part, theoretical upper bounds for are derived using known transitivity properties of the mutual rank probability...
A representation of cyclically ordered sets by means of partial semigroups with an additional unary operation is constructed.