We show that for $\mathrm {C}^*$-algebras with the global Glimm property, the rank of every operator can be realized as the rank of a soft operator, that is, an element whose hereditary sub-$\mathrm {C}^*$-algebra has no nonzero, unital quotients. This implies that the radius of comparison of such a $\mathrm {C}^*$-algebra is determined by the soft part of its Cuntz semigroup.

Under a mild additional assumption, we show that every Cuntz class dominates a (unique) largest soft Cuntz class. This defines a retract from the Cuntz semigroup onto its soft part, and it follows that the covering dimensions of these semigroups differ by at most $1$.

A natural operation on numerical semigroups is taking a quotient by a positive integer. If $\mathcal {S}$ is a quotient of a numerical semigroup with k generators, we call $\mathcal {S}$ a k-quotient. We give a necessary condition for a given numerical semigroup $\mathcal {S}$ to be a k-quotient and present, for each $k \ge 3$, the first known family of numerical semigroups that cannot be written as a k-quotient. We also examine the probability that a randomly selected numerical semigroup with k generators is a k-quotient.

We examine a semigroup analogue of the Kumjian–Renault representation of C*-algebras with Cartan subalgebras on twisted groupoids. Specifically, we represent semigroups with distinguished normal subsemigroups as ‘slice-sections’ of groupoid bundles.

We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

All known structural extensions of the substructural logic $\textbf{FL}_{\textbf{e}}$, the Full Lambek calculus with exchange/commutativity (corresponding to subvarieties of commutative residuated lattices axiomatized by $\{\vee , \cdot , 1\}$-equations), have decidable theoremhood; in particular all the ones defined by knotted axioms enjoy strong decidability properties (such as the finite embeddability property). We provide infinitely many such extensions that have undecidable theoremhood, by encoding machines with undecidable halting problem. An even bigger class of extensions is shown to have undecidable deducibility problem (the corresponding varieties of residuated lattices have undecidable word problem); actually with very few exceptions, such as the knotted axioms and the other prespinal axioms, we prove that undecidability is ubiquitous. Known undecidability results for non-commutative extensions use an encoding that fails in the presence of commutativity, so and-branching counter machines are employed. Even these machines provide encodings that fail to capture proper extensions of commutativity, therefore we introduce a new variant that works on an exponential scale. The correctness of the encoding is established by employing the theory of residuated frames.

We prove that Cuntz semigroups of C*-algebras satisfy Edwards' condition with respect to every quasitrace. This condition is a key ingredient in the study of the realization problem of functions on the cone of quasitraces as ranks of positive elements. In the course of our investigation, we identify additional structure of the Cuntz semigroup of an arbitrary C*-algebra and of the cone of quasitraces.

The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4-element De Morgan monoids C4 and D4, where C4 is the only nontrivial 0-generated algebra onto which finitely subdirectly irreducible De Morgan monoids may be mapped by noninjective homomorphisms. The homomorphic preimages of C4 within DMM (together with the trivial De Morgan monoids) constitute a proper quasivariety, which is shown to have a largest subvariety U. The covers of the variety (C4) within U are revealed here. There are just ten of them (all finitely generated). In exactly six of these ten varieties, all nontrivial members have C4 as a retract. In the varietal join of those six classes, every subquasivariety is a variety—in fact, every finite subdirectly irreducible algebra is projective. Beyond U, all covers of (C4) [or of (D4)] within DMM are discriminator varieties. Of these, we identify infinitely many that are finitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids.

We study comparison properties in the category $\text{Cu}$ aiming to lift results to the ${{\text{C}}^{\text{*}}}$ -algebraic setting. We introduce a new comparison property and relate it to both the corona factorization property $\left( \text{CFP} \right)$ and $\omega $ -comparison. We show differences of all properties by providing examples that suggest that the corona factorization for ${{\text{C}}^{\text{*}}}$ -algebras might allow for both finite and infinite projections. In addition, we show that Rørdam's simple, nuclear ${{\text{C}}^{\text{*}}}$ -algebra with a finite and an inifnite projection does not have the $\text{CFP}$ .

In this paper, we give a new definition of radicals of Green’s relations in an ordered semigroup and characterize left regular (right regular), intra regular ordered semigroups by radicals of Green’s relations. We also characterize the ordered semigroups that are unions and complete semilattices of $\text{t}$ -simple ordered semigroups.

A semiring is a set $S$ with two binary operations $+ $ and $\cdot $ such that both the additive reduct ${S}_{+ } $ and the multiplicative reduct ${S}_{\bullet } $ are semigroups which satisfy the distributive laws. If $R$ is a ring, then, following Chaptal [‘Anneaux dont le demi-groupe multiplicatif est inverse’, C. R. Acad. Sci. Paris Ser. A–B 262 (1966), 274–277], ${R}_{\bullet } $ is a union of groups if and only if ${R}_{\bullet } $ is an inverse semigroup if and only if ${R}_{\bullet } $ is a Clifford semigroup. In Zeleznikow [‘Regular semirings’, Semigroup Forum 23 (1981), 119–136], it is proved that if $R$ is a regular ring then ${R}_{\bullet } $ is orthodox if and only if ${R}_{\bullet } $ is a union of groups if and only if ${R}_{\bullet } $ is an inverse semigroup if and only if ${R}_{\bullet } $ is a Clifford semigroup. The latter result, also known as Zeleznikow’s theorem, does not hold in general even for semirings $S$ with ${S}_{+ } $ a semilattice Zeleznikow [‘Regular semirings’, Semigroup Forum 23 (1981), 119–136]. The Zeleznikow problem on a certain class of semirings involves finding condition(s) such that Zeleznikow’s theorem holds on that class. The main objective of this paper is to solve the Zeleznikow problem for those semirings $S$ for which ${S}_{+ } $ is a semilattice.

We say that a numerical semigroup is $d$ -squashed if it can be written in the form

$$S\,=\,\frac{1}{N}\langle {{a}_{1}},\,.\,.\,.\,,\,{{a}_{d}}\rangle \,\cap \,\mathbb{Z}$$

for $N$ , ${{a}_{1}}\,,\,.\,.\,.\,,\,{{a}_{d}}$ positive integers with $\gcd \left( {{a}_{1}},\,.\,.\,.\,,{{a}_{d}} \right)\,=\,1$ . Rosales and Urbano have shown that a numerical semigroup is 2-squashed if and only if it is proportionally modular.

Recent works by Rosales et al. give a concrete example of a numerical semigroup that cannot be written as an intersection of 2-squashed semigroups. We will show the existence of infinitely many numerical semigroups that cannot be written as an intersection of 2-squashed semigroups. We also will prove the same result for 3-squashed semigroups. We conjecture that there are numerical semigroups that cannot be written as the intersection of $d$ -squashed semigroups for any fixed $d$ , and we prove some partial results towards this conjecture.

A natural topology on the space of left orderings of an arbitrary semi-group is introduced here. This space is proved to be compact, and for free abelian groups it is shown to be homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases.

We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of $\sigma $ -unital simple ${{C}^{*}}$ -algebras $A$ with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra $\mathcal{M}(A)$ , is therefore analyzed. In important cases it is shown that, if $A$ has finite scale then the quotient of $\mathcal{M}(A)$ modulo any closed ideal $I$ that properly contains $A$ has stable rank one. The intricacy of the ideal structure of $\mathcal{M}(A)$ is reflected in the fact that $\mathcal{M}(A)$ can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.

We prove that a commutative preordered semigroup embeds into the space of all equidecomposability types of subsets of some set equipped with a group action (in short, a full type space) if and only if it satisfies the following axioms: (i) (⩝x,y) (x ≤ x + y); (ii) (⩝x,y)((x ≤ y and y ≤ x) ⇒ x = y); (iii) (⩝x,y,u, v)((x + u ≤ y + u and u ≤ v) ⩝ x + v ≤ y ≤ v); (iv) (⩝x, u, V)((x + u = u and u ≤ v) ⇒ x + v = v); (v) (⩝x,y)(mx ≤ my ⇒ x ≤ y) (all m ∊ Ν \ {0}). Furthermore, such a structure can always be embedded into a reduced power of the space Τ of nonempty initial segments of + with rational (possibly infinite) endpoints, equipped with the addition defined by and the ordering defined by . As a corollary, the set of all universal formulas of (+, ≤) satisfied by all full type spaces is decidable.