We prove a cancellation theorem for simple refinement monoids satisfying the weak comparability condition, first introduced by K.C. O’Meara in the context of von Neumann regular rings. This result is then applied to von Neumann regular rings and C∗-algebras of real rank zero via the monoid of isomorphism classes of finitely generated projective modules. Introduction Let M be an (abelian) monoid...

Distributive lattices are well known to be precisely those lattices that possess cancellation: x ∨ y = x ∨ z and x ∧ y = x ∧ z imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the 5-element lattices M3 or N5 as sublattices. In this paper we examine cancellation in skew lattices, where the involved objects are in many ways lattice-like, but the operations ∧ and ∨ no l...

Say that a cone is a commutative monoid that is in addition conical, i.e., satisfies x+y=0 ⇒ x=y=0. We show that cones (resp. simple cones) of many kinds order-embed or even embed unitarily into refinement cones (resp. simple refinement cones) of the same kind, satisfying in addition various divisibility conditions. We do this in particular for all cones, or for all separative cones, or for all...

In a commutative, cancellative, atomic monoid M , the elasticity of a non-unit x is defined to be ρ(x) = L(x)/l(x), where L(x) is the supremum of the lengths of factorizations of x into irreducibles and l(x) is the corresponding infimum. The elasticity ρ(M) of M is given as the supremum of the elasticities of the nonzero non-units in the domain. We call ρ(M) accepted if there exists a non-unit ...

Let (M, deg) be a cancellative monoid M equipped with a discrete degree map deg :M →R≥0, and let PM,deg(t) := ∑ u∈M/∼t deg(u) be its generating series, called the growth function of M . We show the inversion formula PM,deg(t) ·NM,deg(t) = 1 where the second factor of the formula is given by NM,deg(t) := 1 + ∑ T∈Tmcm(M,I0)(−1) #J1+···+#Jn−n+1 ∑ ∆∈|T | t , which we call the skew-growth function o...

Let R be a commutative Noetherian ring of dimension d and M cancellative torsion-free seminormal monoid. Then: (1) A type R[d,m,n] P projective A[M]-module rank r≥max {2,d+1}. Then the action E(A[M]⊕P) on (A[M]⊕P) is transitive (2) Assume (R,m,K) regular local containing field k such that either char k=0 or k=p tr-deg K∕

The catenary degree of an element s of a cancellative commutative monoid S is a nonnegative integer measuring the distance between the irreducible factorizations of s. The catenary degree of the monoid S, defined as the supremum over all catenary degrees occurring in S, has been heavily studied as an invariant of nonunique factorization. In this paper, we investigate the set C(S) of catenary de...

We combine the language of monoids with preorders so as to refine some fundamental aspects classical theory factorization and prove an abstract theorem a variety applications. In particular, we obtain generalization, from cancellative Dedekind-finite (commutative or non-commutative) monoids, on "atomic factorizations" that traces back work P.M. Cohn in 1960s; recover D.D. Anderson S. Valdes-Leo...

where both terms consist of two alternating letters and have the same length. First investigated by J.Tits in the late 1960s [2], and then in [3] and [11], these groups remain incompletely understood, with many open questions, including the decidability of the Word Problem in the general case [6]. The only well understood case is the one of spherical type, which is the case when the associated ...