Totally Ordered Commutative Monoids

Abstract

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

On positive commutative tomonoids

“…It is not difficult to check that our definition is consistent with the definition given, e.g., in [4].…”

“…For finitely generated p.c. tomonoids, there is an easy way to choose a canonical representation, based on the unique minimal set of generators; see, e.g., [4]. However, in this paper we will not do so; we keep matters simpler when accepting the indeterminacy.…”

“…As an example, consider the well-known 9-element non-formally integral tomonoid presented in [4]. Let L be generated by the three elements a, b, and c such that…”

“…An interest in totally ordered monoids, or tomonoids as we say in accordance with [4], is present in diverse fields. For instance, in computational mathematics and in particular in connection with Gröbner bases of polynomial ideals, compatible, positive total orders on an N n , called monomial or term orders, play an important role; see, e.g., [2].…”

“…For instance, in computational mathematics and in particular in connection with Gröbner bases of polynomial ideals, compatible, positive total orders on an N n , called monomial or term orders, play an important role; see, e.g., [2]. An early overview of the extensive research on totally ordered semigroups can be found in [6]; a comprehensive treatment of more recent times is [4].…”