Primary decomposition theorem and generalized spectral characterization of graphs

Prime decomposition theorem for arbitrary semigroups: general holonomy decomposition and synthesis theorem

In the past I have been concerned with the possibility of an extension of the primary decomposition of torsion abelian groups to some standing in arbitrary abelian categories satisfying at least the A.B.5 axiom of Grothendieck [7] and having complete subobject- and factor object lattices which are sets. Such a primary decomposition theorem has been proved in such a general setting only under separate hypotheses and conditions.

The question of computing the group complexity of finite semigroups and automata was first posed in K. Krohn and J. Rhodes, Complexity of finite semigroups, Annals of Mathematics (2) 88 (1968), 128–160, motivated by the Prime Decomposition Theorem of K. Krohn and J. Rhodes, Algebraic theory of machines, I: Prime decomposition theorem for finite semigroups and machines, Transactions of the American Mathematical Society 116 (1965), 450–464. Here we provide an effective lower bound for group complexity.

Product expansions

A prime decomposition theorem for grammatical families

Every automaton can be decomposed into a cascade of basic prime automata. This is the Prime Decomposition Theorem by Krohn and Rhodes. Guided by this theory, we propose automata cascades as a structured, modular, way to describe automata as complex systems made of many components, each implementing a specific functionality. Any automaton can serve as a component; using specific components allows for a fine-grained control of the expressivity of the resulting class of automata; using prime automata as components implies specific expressivity guarantees. Moreover, specifying automata as cascades allows for describing the sample complexity of automata in terms of their components. We show that the sample complexity is linear in the number of components and the maximum complexity of a single component, modulo logarithmic factors. This opens to the possibility of learning automata representing large dynamical systems consisting of many parts interacting with each other. It is in sharp contrast with the established understanding of the sample complexity of automata, described in terms of the overall number of states and input letters, which implies that it is only possible to learn automata where the number of states is linear in the amount of data available. Instead our results show that one can learn automata with a number of states that is exponential in the amount of data available.

It is natural to ask if we can improve on the triangular form. In order to do so, it is clearly necessary to find ‘better’ bases for the subspaces that appear as the direct summands (or primary components) in the Primary Decomposition Theorem. So let us take a closer look at nilpotent mappings.

This chapter is crucial for the study of gauges. Let F be a field with valuation v, and let (F h ,v h ) be the Henselization of (F,v). In §4.1 we prove Morandi’s criterion, Th. 4.1, that v extends to a division algebra D with center F if and only if D⊗ F F h is a division algebra. Among the applications are a primary decomposition theorem and an Ostrowski-type defect theorem, Th. 4.3, for valued division algebras. The valuation v is said to be defectless in D if equality holds in the Fundamental Inequality for the extension of v to D. In §4.2 we extend the notion of defectlessness of v to any semisimple F-algebra A by considering the simple components of A⊗ F F h . In §4.3 we prove key classification results for gauges. We show in Th. 4.26 that if S is a central simple F h -algebra, then any v h -gauge on S is an \(\operatorname {\mathit{End}}\)-gauge as in Prop. 3.34. We also show how v-gauges on a semisimple F-algebra A are built from gauges on the simple components A i of A for the extensions of v to Z(A i ). §4.4 is devoted to proving Th. 4.50, which says that a semisimple F-algebra has a v-gauge if and only if v is defectless in A.

We give a short proof of the Cyclic Decomposition Theorem. The proof proceeds by induction on the dimension of the space in the case that the minimal polynomial of the operator has only one irreducible factor and then uses the Primary Decomposition Theorem to treat the general case.

A prime decomposition theorem for θn-curves in S3

(1994). Noether Lasker Primary Decomposition Revisited. The American Mathematical Monthly: Vol. 101, No. 8, pp. 759-768.

Let R be a commutative ring with identity, S ⊆ R be a multiplicative set, and M be an R-module. We say that a submodule N of M with ( N : R M ) ∩ S = ∅ has an S-primary decomposition if it can be written as a finite intersection of S-primary submodules of M. In this paper, first we provide an example of the S-Noetherian module in which a submodule does not have a primary decomposition. Then our main aim of this paper is to establish the existence and uniqueness of S-primary decomposition in S-Noetherian modules as an extension of a classical Lasker-Noether primary decomposition theorem for Noetherian modules.

On the cantor-dedekind property of the Tilson order on categories and graphs

Introduction. In the following all semigroups are of finite order. One semigroup S, is said to divide another semigroup S2, written SlIS2, if S, is a homomorphic image of a subsemigroup of S2. The semidirect product of S2 by Sl, with connecting homomorphism Y, is written S2 X y Sl. See Definition 1.6. A semigroup S is called irreducible if for all finite semigroups S2 and Si and all connecting homomorphisms Y, S I (S2 X Y SJ) implies S I S2 or S I S1. It is shown that S is irreducible if and only if either:

The Prime Decomposition Theorem for finite transformation semigroups of Krohn and Rhodes and the resultant theory of complexity of finite semigroups are the subjects of this survey. All concepts used are defined in the text, and proofs are given if they are accessible without a large amount of preparation. This article is an attempt to lead the reader to the essential ideas of the theory in the shortest possible time and effort.