Monoids and Maximal Codes

In recent years codes that are not Uniquely Decipherable (UD) are been studied partitioning them in classes that localize the ambiguities of the code. A natural question is how we can extend the notion of maximality to codes that are not UD. In this paper we give an answer to this question. To do this we introduce a partial order in the set of submonoids of a monoid showing the existence, in this poset, of maximal elements that we call full monoids. Then a set of generators of a full monoid is, by definition, a maximal code. We show how this definition extends, in a natural way, the existing definition concerning UD codes and we find a characteristic property of a monoid generated by a maximal UD code.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Random walks on semaphore codes and delay de Bruijn semigroups

We develop a new approach to random walks on de Bruijn graphs over the alphabet $A$ through right congruences on $A^k$, defined using the natural right action of $A^+$. A major role is played by special right congruences, which correspond to semaphore codes and allow an easier computation of the hitting time. We show how right congruences can be approximated by special right congruences.Comment: 34 pages; 10 figures; as requested by the journal, the previous version of this paper was divided into two; this version contains Sections 1-8 of version 1; Sections 9-12 will appear as a separate paper with extra material adde

On Congruence Compact Monoids

A universal algebra is called congruence compact if every family of congruence classes with the finite intersection property has a non-empty intersection. This paper determines the structure of all right congruence compact monoids S for which Green’s relations J and H coincide. The results are thus sufficiently general to describe, in particular, all congruence compact commutative monoids and all right congruence compact Clifford inverse monoids

The Booleanization of an inverse semigroup

We prove that the forgetful functor from the category of Boolean inverse semigroups to inverse semigroups with zero has a left adjoint. This left adjoint is what we term the `Booleanization'. We establish the exact connection between the Booleanization of an inverse semigroup and Paterson's universal groupoid of the inverse semigroup and we explicitly compute the Booleanization of the polycyclic inverse monoid $P_{n}$ and demonstrate its affiliation with the Cuntz-Toeplitz algebra.Comment: This is an updated version of the previous paper. Typos where found have been corrected and a new section added that shows how to construct the Booleanization directly from an arbitrary inverse semigroup with zero (without having to use its distributive completion

Representation Theory of Finite Semigroups over Semirings

We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the results for a field. Special attention is paid to the boolean semiring, where we also characterize the simple representations and introduce the beginnings of a character theory

Agents, subsystems, and the conservation of information

Dividing the world into subsystems is an important component of the scientific method. The choice of subsystems, however, is not defined a priori. Typically, it is dictated by experimental capabilities, which may be different for different agents. Here we propose a way to define subsystems in general physical theories, including theories beyond quantum and classical mechanics. Our construction associates every agent A with a subsystem SA, equipped with its set of states and its set of transformations. In quantum theory, this construction accommodates the notion of subsystems as factors of a tensor product Hilbert space, as well as the notion of subsystems associated to a subalgebra of operators. Classical systems can be interpreted as subsystems of quantum systems in different ways, by applying our construction to agents who have access to different sets of operations, including multiphase covariant channels and certain sets of free operations arising in the resource theory of quantum coherence. After illustrating the basic definitions, we restrict our attention to closed systems, that is, systems where all physical transformations act invertibly and where all states can be generated from a fixed initial state. For closed systems, we propose a dynamical definition of pure states, and show that all the states of all subsystems admit a canonical purification. This result extends the purification principle to a broader setting, in which coherent superpositions can be interpreted as purifications of incoherent mixtures.Comment: 31+26 pages, updated version with new results, contribution to Special Issue on Quantum Information and Foundations, Entropy, GM D'Ariano and P Perinotti, ed

On finite Thurston type orderings of braid groups

We prove that for any finite Thurston-type ordering $<_{T}$ on the braid group\ $B_{n}$, the restriction to the positive braid monoid $(B_{n}^{+},<_{T})$ is a\ well-ordered set of order type $\omega^{\omega^{n-2}}$. The proof uses a combi\ natorial description of the ordering $<_{T}$. Our combinatorial description is \ based on a new normal form for positive braids which we call the \C-normal fo\ rm. It can be seen as a generalization of Burckel's normal form and Dehornoy's \ $\Phi$-normal form (alternating normal form).Comment: 25 pages, 2 figures; proof of Theorem 1 is correcte

The Degree of a Finite Set of Words

We generalize the notions of the degree and composition from uniquely decipherable codes to arbitrary finite sets of words. We prove that if X = Y?Z is a composition of finite sets of words with Y complete, then d(X) = d(Y) ? d(Z), where d(T) is the degree of T. We also show that a finite set is synchronizing if and only if its degree equals one. This is done by considering, for an arbitrary finite set X of words, the transition monoid of an automaton recognizing X^* with multiplicities. We prove a number of results for such monoids, which generalize corresponding results for unambiguous monoids of relations