Abstract. A totally ordered monoid-or tomonoid , for short-is a commutative semigroup with identity S equipped with a total order ≤ S that is translation invariant, i.e., that satisfies: ∀x, y, z ∈ S x ≤ S y ⇒ x + z ≤ S y + z. We call a tomonoid that is a quotient of some totally ordered free commutative monoid formally integral. Our most significant results concern characterizations of this condition by means of constructions in the lattice Z n that are reminiscent of the geometric interpretation of the Buchberger algorithm that occurs in integer programming. In particular, we show that every two-generator tomonoid is formally integral. In addition, we give several (new) examples of tomonoids that are not formally integral, we present results on the structure of nil tomonoids and we show how a valuation-theoretic construction due to Hion reveals relationships between formally integral tomonoids and ordered commutative rings satisfying a condition introduced by Henriksen and Isbell. Introduction.In his 1976 survey of ordered semigroups [G], E. Ya. Gabovich identified several general research problems. The present work contains results that respond directly to at least three of these. First, Gabovich asked explicitly for criteria for formal integrality (and related properties in possibly non-commutative varieties of semigroups). Second, he posed the general problem of developing structure theories for classes of ordered semigroups. Third, he singled out Hion's work as a potentially useful way of describing the structure of totally ordered rings. We now comment on each of these topics in more detail.For clarity, we define a few terms that will be used in the introduction. A monoid is a set with an associative binary operation and an identity element. All monoids in this paper are commutative. The concept of a tomonoid is defined in the abstract. Let S be a tomonoid. We say S is positive if 0 S ≤ S x for all x ∈ S. If S is positive, we say S is archimedean if for any x, y ∈ S \ {0 S } there is a positive integer n such that x ≤ S ny. When we speak of a quotient of S, we always intend a congruence relation θ with convex classes, so the natural surjection S → → S/θ induces a translation-invariant total order on S/θ. We view S/θ as a tomonoid with this order. When we speak of a sub-tomonoid of S, we intend a sub-monoid with induced order.Problem 6 of [G] reads: In a variety (of semigroups) whose free semigroups are orderable, find necessary and sufficient conditions for a totally ordered semigroup to be a quotient of some totally ordered free semigroup. All free commutative monoids are orderableindeed the positive orders on finitely generated free commutative monoids are the so-called 1

For certain classes of negative definite symbols q(x, ξ) and state space dependent Bernstein function f (x, s) we prove that −p(x, D), the pseudo-differential operator with symbol −p(x, ξ) = −f (x, q(x, ξ)), extends to the generator of a Feller semigroup. Our result extends previously known results related to operators of variable (fractional) order of differentiation, or variable order fractional powers. New concrete examples are given.

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Operators induced by Q‐matrices on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$l_p(\mathbb {Z}^m)$\end{document} are shown to satisfy the positive maximum principle and to have a representation as a pseudo‐differential operator with symbol \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$q:\mathbb {Z}^m\times T^m\rightarrow \mathbb {C}$\end{document} which is with respect to the co‐variable negative definite (in the sense of Schoenberg). This observation leads already towards a more geometric interpretation of the transition matrix of the associated Markov chain.

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