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MyhillNerode Theorem for Sequential Transducers over Unique GCDMonoids
"... Abstract. We generalize the classical MyhillNerode theorem for finite automata to the setting of sequential transducers over unique GCDmonoids, which are cancellative monoids in which every two nonzero elements admit a unique greatest common (left) divisor. We prove that a given formal power serie ..."
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Abstract. We generalize the classical MyhillNerode theorem for finite automata to the setting of sequential transducers over unique GCDmonoids, which are cancellative monoids in which every two nonzero elements admit a unique greatest common (left) divisor. We prove that a given formal power
INSIDE FACTORIAL MONOIDS AND INTEGRAL DOMAINS
"... We investigate two classes of monoids and integral domains, called inside and outside factorial, whose definitions are closely related in a divisortheoretic manner to the concept of unique factorization. We prove that a monoid is outside factorial if and only if it is a Krull monoid with torsion cl ..."
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We investigate two classes of monoids and integral domains, called inside and outside factorial, whose definitions are closely related in a divisortheoretic manner to the concept of unique factorization. We prove that a monoid is outside factorial if and only if it is a Krull monoid with torsion
On vdomains: a survey
, 2009
"... An integral domain D is a v–domain if, for every finitely generated nonzero (fractional) ideal F of D, we have (FF −1) −1 = D. The v–domains generalize Prüfer and Krull domains and have appeared in the literature with different names. This paper is the result of an effort to put together information ..."
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An integral domain D is a v–domain if, for every finitely generated nonzero (fractional) ideal F of D, we have (FF −1) −1 = D. The v–domains generalize Prüfer and Krull domains and have appeared in the literature with different names. This paper is the result of an effort to put together information on this useful class of integral domains. In this survey, we present old, recent and new characterizations of v–domains along with some historical remarks. We also discuss the relationship of v–domains with their various specializations and generalizations, giving suitable examples.
GCDSETS IN INTEGRAL DOMAINS
"... Abstract. Let R be an integral domain. A saturated multiplicative subset S 6 = U(R) of R is a GCDset if gcd(a, b) exists for each a, b ∈ S. We study the structure of GCDsets of R, with emphasis on the case where R is a Dedekind domain. We show that if R is atomic, then each GCDset is generated by ..."
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Abstract. Let R be an integral domain. A saturated multiplicative subset S 6 = U(R) of R is a GCDset if gcd(a, b) exists for each a, b ∈ S. We study the structure of GCDsets of R, with emphasis on the case where R is a Dedekind domain. We show that if R is atomic, then each GCDset is generated by completely irreducible elements, and that if R is a Dedekind domain and x is a nonzero nonunit of R, then for some N ≥ 1, xN has a completely irreducible factor. Let R be a Dedekind domain with torsion realizable pair {Cl(R),A}. If S is a GCDset of R, then there is a subgroup GS of Cl(R) generated by an independent subset of A with Cl(R)/GS ∼= Cl(RS). Conversely, suppose that G is a subgroup of Cl(R) generated by an independent subset of A. Then there is a GCDset SG of R with Cl(R)/G ∼= Cl(RSG). 1.