ON SEPARABILITY FINITENESS CONDITIONS IN SEMIGROUPS

Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every $\mathcal {H}$ -class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many $\mathcal {H}$ -classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.

Varieties of commutative semigroups closed under dominions

In this paper, we find sufficient conditions for some commutative varieties of semigroups to be closed under dominions.

ON WEAKLY PERFECT ANNIHILATING-IDEAL GRAPHS

We prove that the annihilating-ideal graph of a commutative semigroup with unity is, in general, not weakly perfect. This settles the conjecture of DeMeyer and Schneider [‘The annihilating-ideal graph of commutative semigroups’, J. Algebra469 (2017), 402–420]. Further, we prove that the zero-divisor graphs of semigroups with respect to semiprime ideals are weakly perfect. This enables us to produce a large class of examples of weakly perfect zero-divisor graphs from a fixed semigroup by choosing different semiprime ideals.

A characterization of Commutative Semigroups

This paper deals with some results on commutative semigroups. We consider (s,.) is externally commutative right zero semigroup is regular if it is intra regular and (s,.) is externally commutative semigroup then every inverse semigroup is u – inverse semigroup. We will also prove that if (S,.) is a H - semigroup then weakly cancellative laws hold in H - semigroup. In one case we will take (S,.) is commutative left regular semi group and we will prove that (S,.) is ∏ - inverse semigroup. We will also consider (S,.) is commutative weakly balanced semigroup and then prove every left (right) regular semigroup is weakly separate, quasi separate and separate. Additionally, if (S,.) is completely regular semigroup we will prove that (S,.) is permutable and weakly separtive. One a conclusing note we will show and prove some theorems related to permutable semigroups and GC commutative Semigroups.

Commutative semigroups whose endomorphisms are power functions

For any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

On dominions and varieties of bands

We have shown that a variety of left(right) normal bands is closed in some homotypical varieties. Further, we partially generalize a result of Isbell from the class of commutative semigroups to some generalized classes of commutative semigroups by showing that dominion of such semigroups belongs to same classes.

A linear-time algorithm that avoids inverses and computes Jackknife (leave-one-out) products like convolutions or other operators in commutative semigroups

Background Data about herpesvirus microRNA motifs on human circular RNAs suggested the following statistical question. Consider independent random counts, not necessarily identically distributed. Conditioned on the sum, decide whether one of the counts is unusually large. Exact computation of the p-value leads to a specific algorithmic problem. Given $$n$$ n elements $$g_{0} ,g_{1} , \ldots ,g_{n - 1}$$ g 0 , g 1 , … , g n - 1 in a set $$G$$ G with the closure and associative properties and a commutative product without inverses, compute the jackknife (leave-one-out) products $$\bar{g}_{j} = g_{0} g_{1} \cdots g_{j - 1} g_{j + 1} \cdots g_{n - 1}$$ g ¯ j = g 0 g 1 ⋯ g j - 1 g j + 1 ⋯ g n - 1 ($$0 \le j < n$$ 0 ≤ j < n ). Results This article gives a linear-time Jackknife Product algorithm. Its upward phase constructs a standard segment tree for computing segment products like $$g_{{\left[ {i,j} \right)}} = g_{i} g_{i + 1} \cdots g_{j - 1}$$ g i , j = g i g i + 1 ⋯ g j - 1 ; its novel downward phase mirrors the upward phase while exploiting the symmetry of $$g_{j}$$ g j and its complement $$\bar{g}_{j}$$ g ¯ j . The algorithm requires storage for $$2n$$ 2 n elements of $$G$$ G and only about $$3n$$ 3 n products. In contrast, the standard segment tree algorithms require about $$n$$ n products for construction and $$\log_{2} n$$ log 2 n products for calculating each $$\bar{g}_{j}$$ g ¯ j , i.e., about $$n\log_{2} n$$ n log 2 n products in total; and a naïve quadratic algorithm using $$n - 2$$ n - 2 element-by-element products to compute each $$\bar{g}_{j}$$ g ¯ j requires $$n\left( {n - 2} \right)$$ n n - 2 products. Conclusions In the herpesvirus application, the Jackknife Product algorithm required 15 min; standard segment tree algorithms would have taken an estimated 3 h; and the quadratic algorithm, an estimated 1 month. The Jackknife Product algorithm has many possible uses in bioinformatics and statistics.

A linear-time algorithm that avoids inverses and computes jackknife (leave-one-out) products like convolutions or other operators in commutative semigroups

[Please see the manuscript file pdf to view the full abstract.]Background: Data about herpesvirus microRNA motifs on human circular RNAs suggested the following statistical question. Consider independent random counts, not necessarily identically distributed. Conditioned on the sum, decide whether one of the counts is unusually large. Exact computation of the p-value leads to a specific algorithmic problem. Given elements in a set with the closure and associative properties and a commutative product without inverses, compute the jackknife (leave-one-out) products ( ).Results: This article gives a linear-time Jackknife Product algorithm. Its upward phase constructs a standard segment tree for computing segment products like ; its novel downward phase mirrors the upward phase while exploiting the symmetry of and its complement . The algorithm requires storage for elements of and only about products. In contrast, the standard segment tree algorithms require about products for construction and products for calculating each , i.e., about products in total; and a naïve quadratic algorithm using element-by-element products to compute each requires products.Conclusions: In the herpesvirus application, the Jackknife Product algorithm required 15 minutes; standard segment tree algorithms would have taken an estimated 3 hours; and the quadratic algorithm, an estimated 1 month. The Jackknife Product algorithm has many possible uses in bioinformatics and statistics.