scholarly journals Learning Weighted Automata over Principal Ideal Domains

2020 ◽  
pp. 602-621 ◽  
Author(s):  
Gerco van Heerdt ◽  
Clemens Kupke ◽  
Jurriaan Rot ◽  
Alexandra Silva
Keyword(s):  
Active Learning ◽  
Principal Ideal ◽  
Learning Algorithms ◽  
Principal Ideal Domain ◽  
Natural Numbers ◽  
Ideal Domain ◽  
Weighted Automata ◽  
Principal Ideal Domains

AbstractIn this paper, we study active learning algorithms for weighted automata over a semiring. We show that a variant of Angluin’s seminal $$\mathtt {L}^{\!\star }$$ L ⋆ algorithm works when the semiring is a principal ideal domain, but not for general semirings such as the natural numbers.

S-∗w-Principal Ideal Domains

2018 ◽  
Vol 25 (02) ◽  
pp. 217-224 ◽  
Author(s):  
Hwankoo Kim ◽  
Jung Wook Lim
Keyword(s):  
Principal Ideal ◽  
Integral Domain ◽  
Type Theorem ◽  
Local Property ◽  
Nonzero Ideal ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
Star Operation ◽  
Principal Ideal Domains

Let D be an integral domain, ∗ a star-operation on D, and S a multiplicative subset of D. We define D to be an S-∗w-principal ideal domain if for each nonzero ideal I of D, there exist an element s ∈ S and a principal ideal (c) of D such that [Formula: see text]. In this paper, we study some properties of S-∗w-principal ideal domains. Among other things, we study the local property, the Nagata type theorem, and the Cohen type theorem for S-∗w-principal ideal domains.

scholarly journals Notes on topological rings

2013 ◽  
Vol 29 (2) ◽  
pp. 267-273
Author(s):  
MIHAIL URSUL ◽  
◽  
MARTIN JURAS ◽  
Keyword(s):  
Principal Ideal ◽  
Principal Ideal Domain ◽  
Compact Ring ◽  
Ring Topology ◽  
Bezout Domain ◽  
Totally Bounded ◽  
Ideal Domain ◽  
Nilpotent Ring ◽  
Trivial Multiplication

We prove that every infinite nilpotent ring R admits a ring topology T for which (R, T ) has an open totally bounded countable subring with trivial multiplication. A new example of a compact ring R for which R2 is not closed, is given. We prove that every compact Bezout domain is a principal ideal domain.

Invariant Factors Under Rank One Perturbations

1980 ◽  
Vol 32 (1) ◽  
pp. 240-245 ◽  
Author(s):  
Robert C. Thompson
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Matrix Multiplication ◽  
Diagonal Form ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
Zero Divisors ◽  
Rank One ◽  
Invariant Factors

Let R be a principal ideal domain, i.e., a commutative ring without zero divisors in which every ideal is principal. The invariant factors of a matrix A with entries in R are the diagonal elements when A is converted to a diagonal form D = UAV, where U, V have entries in R and are unimodular (invertible over R), and the diagonal entries d1 …, dn of D form a divisibility chain: d1|d2| … |dn. Very little has been proved about how invariant factors may change when matrices are added. This is in contrast to the corresponding question for matrix multiplication, where much information is now available [6].

scholarly journals Finitely generated cyclic extensions of free groups are residually finite

1971 ◽  
Vol 5 (1) ◽  
pp. 87-94 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Free Group ◽  
Principal Ideal ◽  
Free Groups ◽  
Cyclic Extension ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Residually Finite ◽  
Ideal Domain ◽  
Direct Sum

We establish the result that a finitely generated cyclic extension of a free group is residually finite. This is done, in part, by making use of the fact that a finitely generated module over a principal ideal domain is a direct sum of cyclic modules.

scholarly journals Decomposition of representation algebras

1969 ◽  
Vol 10 (3-4) ◽  
pp. 395-402 ◽  
Author(s):  
W. D. Wallis
Keyword(s):  
Finite Group ◽  
Principal Ideal ◽  
Isomorphism Class ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
A Finite Group

Throughout this paper g is a finite group and f is a complete local principal ideal domain of characteristic p where p divides |g|. The notations of [5] are adopted; moreover we shall denote the isomorphism-class of an f g-representation module ℳ by M, the class of ℳx by Mx and the class of ℳR by MR for suitable groups K and R.

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