Elastic Properties and Prime Elements

Abstract

Abstract. In a commutative, cancellative, atomic monoid M , the elasticity of a non-unit x is defined to be ρ(x) = L(x)/l(x), where L(x) is the supremum of the lengths of factorizations of x into irreducibles and l(x) is the corresponding infimum. The elasticity ρ(M ) of M is given as the supremum of the elasticities of the nonzero non-units in the domain. We call ρ(M ) accepted if there exists a non-unit x ∈ M with ρ(M ) = ρ(x). In this paper, we show for a monoid M with accepted elasticity thatif M has a prime element. We develop the ideas of taut and flexible elements to study the set {ρ(x) | x a non-unit of M } when M does not possess a prime element. Mathematics Subject Classification (2000). 20M14, 13F20, 13F15.

A theorem on accepted elasticity in certain local arithmetical congruence monoids

“…This implies that each A-atom in the factorization of W must have exactly two prime factors in [3]. However, as having a factor in [3] means it can have no prime factors in [2] (as [3] + [2] = [1]), each A-type atom must be of the form p 2k q 1 q 2 r 1 . .…”

“…Each prime factor other than p which divides an A-type atom must be in [3], [2], or [0] (this holds as every prime in [1] has value 3). Moreover, each B-type atom must have at least one element in [1] or [3] (as p k is in [3], any number with p-adic value k with no prime factor other than p in [1] or [3] must be in either [1] or [3] itself and therefore not in M(p k a, p k b)).…”

“…Letting r = n/ϕ(b), we observe that as n/r and n must be even, k must be odd; in addition, the maximum p-adic value of an atom is k + (k + 1) − 1 = 2k. Letting s be a prime in [n/r − 1] and h be a prime in [2], we can let x = (p 2k s k−1−(r−1)n/r ) k−1 (p 2k h 2n 2 +1 ); as (p 2k s k−1−(r−1)n/r ) and p 2k have no factors of the form p k w where w ∈ [1] and w is not divisible by p (all elements of the monoid of padic value k must be of this form, and if an element of p-adic value 2k can be factored in the monoid it must be expressible as the product of two elements of p-adic value k), they are all atoms so x can be written as the product of k atoms. However, x is also equal to (p k sh) 2k−1 (p k s (k−1)(k−1−(r−1)n/r)−(2k−1) h 2n 2 −2k+2 ); p k sh, being of minimal p-adic value, is clearly an atom.…”

Systems of Sets of Lengths: Transfer Krull Monoids Versus Weakly Krull Monoids

“…Let If k ∈ {1, 3}, then the sets [2,3], [3,6] shows that L(U 1 U 2 ) = [2,4]. It remains to verify the following assertions.…”

“…We use the same notation as in A1, and assert that L(U 14] which implies that [6,14] ∈ L(G). Suppose that k ≥ 9, and that the assertion holds for all…”