Identities in ordered sets (Q1304914)

The paper deals with lattice-like features of ordered sets. The author introduces the notions of term and identity for ordered sets in the language using, besides the lattice operations, also the symbols \(L\) and \(U\) which are interpreted as the sets of all lower and upper bounds, respectively, of the unions of arguments. It is shown that the identities of ordered sets are in a one-one correspondence with the lattice identities. In connection with this result a question arises: which lattice identities are inherited from any ordered set to its characteristic lattice (i.e., to the lattice generated by an ordered set in its Dedekind-MacNeille completion). The author shows that this is valid for the distributive identity but it fails in the case of the modular identity.