Abstract not found

A rigorous language for discussing the issue of configuration management is currently lacking. To this end, we develop a simple state-machine model of configuration management. Observed behaviors comprise the state of a host and configuration processes accomplish state transitions. Using this language, we show that for one host in isolation and for some configuration processes, reproducibility of observed effect for a configuration process is a statically verifiable property of the process. Using configuration processes verified in this manner, we can efficiently identify latent preconditions that affect behavior among a population of hosts. Constructing configuration management tools with statically verifiable observed behaviors thus reduces the lifecycle cost of configuration management. 1

Among the most important and intensively studied classes of semigroups are finite semigroups, regular semigroups and inverse semigroups. Finite semigroups arise as syntactic semigroups of regular languages and as transition semigroups of finite automata. This connection has lead

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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. 0.

Contents Introduction 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Extensions 5 1.1 Extensions and expansions . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Factor systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Abelian extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Quandle extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5 Involutory extensions . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Modules 29 2.1 Rack modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Beck modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 A digression on G-modules . . . . . . . . . . . . . . . . . . . . . 41 2.4 Free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 The rack

Using techniques of Rewriting Theory, we present a new proof of the known theorem of Munn that FIX , the free inverse semigroup on X , is isomorphic to birooted word-trees on X . Inverse semigroups form a variety of semigroups with one additional unary operation, ( ) \Gamma1 . The following set I of equations axiomatizes it (see [1]): x(yz) = (xy)z (1) (x \Gamma1 ) \Gamma1 = x (2) (xy) \Gamma1 = y \Gamma1 x \Gamma1 (3) xx \Gamma1 x = x (4) xx \Gamma1 yy \Gamma1 = yy \Gamma1 xx \Gamma1 : (5) The set of terms over X , TX , is defined recursively by: (1) Every x 2 X is a term, (2) If t 1 ; t 2 are terms, then t 1 t 2 and (t 1 ) \Gamma1 are terms. We will avoid superfluous parentheses, e.g. will write xyz instead of (xy)z or x(yz). Note that for each term t 2 TX , there is an I -equivalent term w t = u 1 \Delta \Delta \Delta u n , with u i 2 X [ fa \Gamma1 : a 2 Xg (apply repeatedly Ax. (2) and (3)). For a given set X , denote by GX the set of finite, direct...

The concept of a fundamental semigroup, which is a semigroup that cannot be shrunk homomorphically without collapsing idempotents together, was introduced by Munn [25] in 1966, who developed an elegant theory within the class of inverse semigroups, which inspired many researchers in subsequent decades. There is a natural partial order on the

this paper we develop an algorithm which, given a presentation for a commutative semigroup S and the characteristic of a eld k, decides whether the semigroup algebra k[S] is a principal ideal ring with identity. This builds upon the work of Rosales, Garca-Sanchez and Garca-Garca ([14], [15]) who have developed a number of useful algorithms for computing with nitely presented commutative semigroups: most importantly an algorithm to compute the (necessarily nite) set of idempotents. Much of the remaining work depends on nding presentations for ideals, subgroups and quotients of nitely presented commutative semigroups. This draws on techniques developed by Ruskuc and others in [3] and [16] for nding presentations of subsemigroups

The connection between groups and recursive (un)decidability has a long history, going back to the early 1900s. Also, various polynomial-time algorithms have been known in group theory for a long time. However the impact of more general computational complexity (e.g., NP-completeness or PSpace-completeness) has been relatively small and recent. These lectures review a sampling of older facts about algorithmic problems in group theory, and then present more recent results about the connection with complexity: isoperimetric functions and NP; Thompson groups, boolean circuits, and coNP; Thompson monoids and circuit complexity; Thompson groups, reversible computing, and #P; distortion of Thompson groups within Thompson monoids, and one-way permutations. We are especially interested in deep connections between computational complexity and group theory. By “connection ” we do not just mean analyzing the computational complexity of algorithms about groups. We are more interested in algebraic characterizations of complexity classes in terms of group theory, i.e., in finding a “mirror image” of all of complexity theory within group theory. Conversely, we are interested in the computational nature of concepts that appear at first purely algebraic.