On congruence compact monoids
نویسندگان
چکیده
منابع مشابه
Arithmetic of Congruence Monoids
Let N represent the positive integers. Let n ∈ N and Γ ⊆ N. Set Γn = {x ∈ N : ∃y ∈ Γ, x ≡ y (mod n)} ∪ {1}. If Γn is closed under multiplication, it is known as a congruence monoid or CM. A classical result of James and Niven [15] is that for each n, exactly one CM admits unique factorization into products of irreducibles, namely Γn = {x ∈ N : gcd(x, n) = 1}. In this paper, we examine additiona...
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A function on an algebra is congruence preserving if, for any congruence, it maps congruent elements to congruent elements. We show that, on a free monoid generated by at least three letters, a function from the free monoid into itself is congruence preserving if and only if it is of the form x 7→ w0xw1 · · ·wn−1xwn for some finite sequence of words w0, . . . , wn. We generalize this result to ...
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Let n ∈ N, Γ ⊆ N and define Γn = {x ∈ Zn | x ∈ Γ} the set of residues of elements of Γ modulo n. If Γn is multiplicatively closed we may define the following submonoid of the naturals: HΓn = {x ∈ N | x = γ, γ ∈ Γn}∪{1} known as a congruence monoid (CM). Unlike the naturals, many CMs enjoy the property of non-unique factorization into irreducibles. This opens the door to the study of arithmetic ...
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We consider multiplicative monoids of the positive integers defined by a single congruence. If a and b are positive integers such that a≤ b and a2≡ a mod b, then such a monoid (known as an arithmetic congruence monoid or an ACM) can be described as Ma,b = (a+ bN0)∪{1}. In lectures on elementary number theory, Hilbert demonstrated to students the utility of the proof of the Fundamental Theorem o...
متن کاملOn the Elasticity of Generalized Arithmetical Congruence Monoids
An arithmetical congruence monoid (or ACM ) is a multiplicative monoid, which consists of an arithmetic sequence and the element 1. As they are traditionally defined, it is required that a ≤ b and a ≡ a (mod b) must hold to ensure closure. It is well known that unique factorization need not occur in ACMs. In this paper, we investigate factorization results when the requirement a ≤ b is dropped....
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ژورنال
عنوان ژورنال: Mathematika
سال: 1999
ISSN: 0025-5793,2041-7942
DOI: 10.1112/s0025579300007701