It is well known that intra-regular semigroups (ordered semigroups) play an essential role in studying the structure of semigroups (ordered semigroups). In this paper we present a structure theorem referring to the decomposition of ordered semigroups into left strongly simple components, that is, into components, which are both simple and left quasi-regular. We prove, among others, that if an o...

The Tamari lattice of order n can be defined on the set Tn of binary trees endowed with the partial order relation induced by the well-known rotation transformation. In this paper, we restrict our attention to the subset Mn of Motzkin trees. This set appears as a filter of the Tamari lattice. We prove that its diameter is 2n − 5 and that its radius is n − 2. Enumeration results are given for jo...

Any finite distributive lattice has a unique Heyting structure. Every finite Heyting semilattice has, of course, a unique lattice structure and it is necessarily distributive. ∗ This ms was born in 2002. The first appendix was added in 2005, the first footnote in 2013, the subscorings in 2015 and the addendum in 2017. 1 [ ] See 2nd appendix for subscorings. First x and y meet x↔ y in the same w...