Monotone Lévy processes with additive increments are defined and studied. It is shown that these processes have natural Markov structure and their Markov transition semigroups are characterized using the monotone Lévy-Khintchine formula. 17 Monotone Lévy processes turn out to be related to classical Lévy processes via Attal's " remarkable transformation. " A monotone analogue of the family of e...

Exactly as in semigroups, Green’s relations play an important role in the theory of ordered semigroups—especially for decompositions of such semigroups. In this paper we deal with the -trivial ordered semigroups which are defined via the Green’s relation , and with the nil and Δ-ordered semigroups. We prove that every nil ordered semigroup is -trivial which means that there is no ordered semigr...

A concept of ideal extensions in ternary semigroups is introduced and throughly investigated. The connection between an ideal extensions and semilattice congruences in ternary semigroups is considered.

Consider an interval I ⊆ Q . Set S(I) = {m ∈ N : ∃ n ∈ N , mn ∈ I}. This turns out to be a numerical semigroup, and has been the subject of considerable recent investigation (see Chapter 4 of [2] for an introduction). Special cases include modular numerical semigroups (see [4]) where I = [mn , m n−1 ] (m,n ∈ N ), proportionally modular numerical semigroups (see [3]) where I = [mn , m n−s ] (m,n...

Among the most important and intensively studied classes of semigroups are finite semigroups, regular semigroups and inverse semigroups. Finite semigroups arise as syntactic semigroups of regular languages and as transition semigroups of finite automata. This connection has lead to a large and deep literature on classifying regular languages by means of algebraic properties of their correspondi...