The monoid consisting of Kuratowski operations

The paper fills gaps in knowledge about Kuratowski operations which are already in the literature. The Cayley table for these operations has been drawn up. Techniques, using only paper and pencil, to point out all semigroups and its isomorphic types are applied. Some results apply only to topology, one can not bring them out, using only properties of the complement and a closure-like operation. The arguments are by systematic study of possibilities.Comment: We are going to submit the article to a journa

Uniform and Bernoulli measures on the boundary of trace monoids

Trace monoids and heaps of pieces appear in various contexts in combinatorics. They also constitute a model used in computer science to describe the executions of asynchronous systems. The design of a natural probabilistic layer on top of the model has been a long standing challenge. The difficulty comes from the presence of commuting pieces and from the absence of a global clock. In this paper, we introduce and study the class of Bernoulli probability measures that we claim to be the simplest adequate probability measures on infinite traces. For this, we strongly rely on the theory of trace combinatorics with the M\"obius polynomial in the key role. These new measures provide a theoretical foundation for the probabilistic study of concurrent systems.Comment: 34 pages, 5 figures, 27 reference

Bianchi spaces and their 3-dimensional isometries as S-expansions of 2-dimensional isometries

In this paper we show that some 3-dimensional isometry algebras, specifically those of type I, II, III and V (according Bianchi's classification), can be obtained as expansions of the isometries in 2 dimensions. It is shown that in general more than one semigroup will lead to the same result. It is impossible to obtain the algebras of type IV, VI-IX as an expansion from the isometry algebras in 2 dimensions. This means that the first set of algebras has properties that can be obtained from isometries in 2 dimensions while the second set has properties that are in some sense intrinsic in 3 dimensions. All the results are checked with computer programs. This procedure can be generalized to higher dimensions, which could be useful for diverse physical applications.Comment: 23 pages, one of the authors is new, title corrected, finite semigroup programming is added, the semigroup construction procedure is checked by computer programs, references to semigroup programming are added, last section is extended, appendix added, discussion of all the types of Bianchi spaces is include

Monoids of modules and arithmetic of direct-sum decompositions

Let $R$ be a (possibly noncommutative) ring and let $\mathcal C$ be a class of finitely generated (right) $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set $\mathcal V (\mathcal C)$ of isomorphism classes of modules is a commutative semigroup with operation induced by the direct sum. This semigroup encodes all possible information about direct sum decompositions of modules in $\mathcal C$. If the endomorphism ring of each module in $\mathcal C$ is semilocal, then $\mathcal V (\mathcal C)$ is a Krull monoid. Although this fact was observed nearly a decade ago, the focus of study thus far has been on ring- and module-theoretic conditions enforcing that $\mathcal V(\mathcal C)$ is Krull. If $\mathcal V(\mathcal C)$ is Krull, its arithmetic depends only on the class group of $\mathcal V(\mathcal C)$ and the set of classes containing prime divisors. In this paper we provide the first systematic treatment to study the direct-sum decompositions of modules using methods from Factorization Theory of Krull monoids. We do this when $\mathcal C$ is the class of finitely generated torsion-free modules over certain one- and two-dimensional commutative Noetherian local rings.Comment: Pacific Journal of Mathematics, to appea

Classification of grouplike categories

In this paper we study grouplike monoids, these are monoids that contain a group to which we add an ordered set of idempotents. We classify finite categories with two objects having grouplike endomorphism monoids, and we give a count of certain categories with grouplike monoids.Comment: Minor changes in Lemma 3.18. Added Definition 4.3, Lemma 4.4 and Remark 4.5. And the proof of Proposition 4.6 is improve

Quotients by actions of the derived group of a maximal unipotent subgroup

Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. If $X$ is an affine $G$-variety, then the algebra of $U'$-invariants, k[X]^U', is finitely generated and the quotient morphism $\pi: X \to X//U'$ is well-defined. In this article, we study properties of such quotient morphisms, e.g. the property that all the fibres of $\pi$ are equidimensional. We also establish an analogue of the Hilbert-Mumford criterion for the null-cones with respect to $U'$-invariants.Comment: 23 pages, final version, to appear in Pacific J Mat

Two polygraphic presentations of Petri nets

This document gives an algebraic and two polygraphic translations of Petri nets, all three providing an easier way to describe reductions and to identify some of them. The first one sees places as generators of a commutative monoid and transitions as rewriting rules on it: this setting is totally equivalent to Petri nets, but lacks any graphical intuition. The second one considers places as 1-dimensional cells and transitions as 2-dimensional ones: this translation recovers a graphical meaning but raises many difficulties since it uses explicit permutations. Finally, the third translation sees places as degenerated 2-dimensional cells and transitions as 3-dimensional ones: this is a setting equivalent to Petri nets, equipped with a graphical interpretation.Comment: 28 pages, 24 figure