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Attila Nagy - One of the best experts on this subject based on the ideXlab platform.

  • On Monoid Congruences of Commutative Semigroups
    arXiv: Group Theory, 2015
    Co-Authors: Attila Nagy
    Abstract:

    In this paper we characterize the monoid congruences of Commutative Semigroups by the help of the notion of the separator of subsets of Semigroups. We show that every monoid congruence of a Commutative Semigroup S can be constructed by the help of subsets A of S whose separator is not empty.

  • Weakly separative weakly Commutative Semigroups
    Semigroup Forum, 2014
    Co-Authors: Attila Nagy
    Abstract:

    A Semigroup S is called a weakly Commutative Semigroup if, for every a,b∈S, there is a positive integer n such that (ab) n ∈Sa∩bS. A Semigroup S is called archimedean if, for every a,b∈S, there are positive integers m and n such that a n ∈SbS and b m ∈SaS. It is known that every weakly Commutative Semigroup is a semilattice of weakly Commutative archimedean Semigroups. A Semigroup S is called a weakly separative Semigroup if, for every a,b∈S, the assumption a 2=ab=b 2 implies a=b. In this paper we show that a weakly Commutative Semigroup is weakly separative if and only if its archimedean components are weakly cancellative. This result is a generalization of Theorem 4.16 of Clifford and Preston (The Algebraic Theory of Semigroups, Am. Math. Soc., Providence, 1961).

  • Subdirectly irreducible $\mathcal{RGC}$-Commutative right H-Semigroups
    Semigroup Forum, 2008
    Co-Authors: Attila Nagy
    Abstract:

    A Semigroup S is said to be ℛ-Commutative if, for all elements a,b∈S, there is an element x∈S1 such that ab=bax. A Semigroup S is called a generalized conditionally Commutative (briefly, \(\mathcal{GC}\) -Commutative) Semigroup if it satisfies the identity aba2=a2ba. An ℛ-Commutative and \(\mathcal{GC}\) -Commutative Semigroup is called an \(\mathcal{RGC}\) -Commutative Semigroup. A Semigroup S is said to be a right H-Semigroup if every right congruence of S is a congruence of S. In this paper we characterize the subdirectly irreducible Semigroups in the class of \(\mathcal{RGC}\) -Commutative right H-Semigroups.

  • (m, n)-Commutative Semigroups
    Special Classes of Semigroups, 2001
    Co-Authors: Attila Nagy
    Abstract:

    In the last two chapters we deal with the (m, n)-Commutative and the n (2) permutable Semigroups, respectively. A Semigroup is called an (m, n)-Commutative Semigroup if it satisfies the identity (x 1 ... x m )(y 1 ... y n ) (m and n are positive i n tegers). For a fixed integer n ≥ 2, a Semigroup S is cailed an n (2)-permu table Semigroup if, for any n-tuple (x 1, x 2, ..., x n ) of elements of S, there is a positive in teger t with 1 ≤ t ≤ n-1 such that x 1 x 2 ... x t x t + 1 ... x n = x t + 1... x n x 1...x t . First we deat with the (m, n)-co mmutative Semigroups, because some results about them are necessary in the ex a m ina tio ns of n(2)- per mutable ones. In this chapter the (m, n)-Commutative Semigroups are examined. In the first part of the chapter we determine all couples (m, n) of positive integers m and n for which a Semigroup is (m, n)-Commutative. Since an (m, n)Commutative Semigroup S is (m′, n′)-Commutative for every m′ ≥ m and n′ ≥ n, it is sufficient to know the function fs(n) = min{m : S is (m, n) — Commutative}. As every (m, n)-Commutative Semigroup is (1, m + n)-Commutative, fs is defined for all positive integers. We define a special function, the permutation function, and show that the functions Is are exactly the permutation functions. In the second part of the chapter, we show that every (m, n)-Commutative Semigroup is an E-k Semigroup for some integer k ≥ Z. We also show that every (1, 2)-Commutative Semigroup is exponential. In the third part of the chapter, we deal with the semilattice decomposition of (m, n)-Commutative Semigroups.

  • Weakly Commutative Semigroups
    Special Classes of Semigroups, 2001
    Co-Authors: Attila Nagy
    Abstract:

    A Semigroup S is called left (right) weakly Commutative if, for every a, b ∈ S, there exist x ∈ S and a positive integer n such that (ab) n = bx ((ab) n = xa). A Semigroup which is both left and right weakly Commutative is called a weakly Commutative Semigroup. In this chapter we deal with left weakly Commutative, right weakly Commutative and weakly Commutative Semigroups. It is shown that a Semigroup is a semilattice of left archimedean (right archimedean, t-archimedean) Semigroups if and only if it is right weakly Commutative (left weakly Commutative, weakly Commutative). It is proved that a weakly Commutative 0-simple Semigroup is a group with a zero adjoined. Moreover, a Semigroup is weakly Commutative archimedean and contains an idempotent element if and only if it is an ideal extension of a group by a nil Semigroup. We get, as a consequence, that a Semigroup is weakly Commutative and regular if and only if it is a Clifford Semigroup. We show that a right (left) weakly Commutative Semigroup is embeddable into a group if and only if it is cancellative. At the end of the chapter we deal with the least weakly separative congruence on weakly Commutative Semigroups. It is proved that if 5 is a left weakly Commutative Semigroup then σ defined by a σ b if and only if ab n = bn+1 and ba n = an+1 for a positive integer n is a weakly separative congruence on 5. Similarly, if S is a right weakly Commutative Semigroup then r defined by a τ b if and only if b n a = b n+1 and a n b = an+1 for some positive integer n is a weakly separative congruence on S. Moreover, π = σ ∩ τ is the least weakly separative congruence on a weakly Commutative Semigroup.

Zhang Yuanyuan - One of the best experts on this subject based on the ideXlab platform.

  • Free pre-Lie family algebras
    HAL CCSD, 2020
    Co-Authors: Manchon Dominique, Zhang Yuanyuan
    Abstract:

    24 pages, tikz figuresIn this paper, we first define the pre-Lie family algebra associated to a dendriform family algebra in the case of a Commutative Semigroup. Then we construct a pre-Lie family algebra via typed decorated rooted trees, and we prove the freeness of this pre-Lie family algebra. We also construct pre-Lie family operad in terms of typed labeled rooted trees, and we obtain that the operad of pre-Lie family algebras is isomorphic to the operad of typed labeled rooted trees, which generalizes the result of F. Chapoton and M. Livernet. In the end, we construct Zinbiel and pre-Poisson family algebras and generalize results of M. Aguiar

  • Free pre-Lie family algebras
    2020
    Co-Authors: Manchon Dominique, Zhang Yuanyuan
    Abstract:

    In this paper, we first define the pre-Lie family algebra associated to a dendriform family algebra in the case of a Commutative Semigroup. Then we construct a pre-Lie family algebra via typed decorated rooted trees, and we prove the freeness of this pre-Lie family algebra. We also construct pre-Lie family operad in terms of typed labeled rooted trees, and we obtain that the operad of pre-Lie family algebras is isomorphic to the operad of typed labeled rooted trees, which generalizes the result of F. Chapoton and M. Livernet. In the end, we construct Zinbiel and pre-Poisson family algebras and generalize results of M. Aguiar.Comment: 24 pages, tikz figure

A V Kelarev - One of the best experts on this subject based on the ideXlab platform.

  • an algorithm for Commutative Semigroup algebras which are principal ideal rings
    Communications in Algebra, 2004
    Co-Authors: I M Araujo, A V Kelarev, A Solomon
    Abstract:

    Abstract We outline several algorithms for computing with Semigroup representations, and combine them in order to verify when certain algebras are principal ideal rings.

  • the jacobson radical of Commutative Semigroup rings
    Journal of Algebra, 1992
    Co-Authors: A V Kelarev
    Abstract:

    In this paper we consider semiprimitive Commutative Semigroup rings and related matters. A ring is said to be semiprhnitive if the Jacobson radical of it is equal to zero. This property is one of the most important in the theory of Semigroup rings, and there is a prolific literature pertaining to the field (see 1,,14]). All semiprimitive rings are contained in another interesting class of rings. Let 8 denote the class of rings R such that ~/(R) = B(R), where J and B are the Jacobson and Baer radicals. Clearly, every semiprimitive ring is in 6". This class, appears, for example, in the theory of Pl-rings and in Commutative algebra. (In particular, every finitely generated PI-ring and every Hilbert ring are in 6".) Therefore, it is of an independent interest. Meanwhile it is all the more interesting because any characterization of the Semigroup rings in 6" will immediately give us a description of semiprimitive Semigroup rings. Indeed, a ring R is semiprimitive if and only if R~6" and R is semiprime, i.e., B(R)=O. Semiprime Commutative Semigroup rings have been described by Parker and Giimer 1,12] and, in other terms, by Munn [9]. So it suffices to characterize Semigroup rings in 6". Semigroup rings of 6" were considered by Karpilovsky r5], Munn 1,6-9], Okninski 1-10-h and others. In this paper Commutative Semigroup rings which are in 6" will be described completely. To this end one should know the structure of the Jacobson radical J(R[S]). In [2] Jespers described J(R1,S]) under rather weak assumptions on R. They hold, in particular, for every Commutative R. Here we shall give another (quite short) description of J(R1,S]) which does not require any restriction on R. Besides, it is specially fitted for testing whether an element is in J(R1,S]), and this is essential for our proofs.

  • The regular radical of Semigroup rings of Commutative Semigroups
    Glasgow Mathematical Journal, 1992
    Co-Authors: A V Kelarev
    Abstract:

    A description of regular group rings is well known (see [12]). Various authors have considered regular Semigroup rings (see [17], [8], [10], [11], [4]). These rings have been characterized for many important classes of Semigroups, although the general problem turns out to be rather difficult and still has not got a complete solution. It seems natural to describe the regular radical in Semigroup rings for Semigroups of the classes mentioned. In [10], the regular Semigroup rings of Commutative Semigroups were described. The aim of the present paper is to characterize the regular radical ρ(R[S]) for each associative ring R and Commutative Semigroup S.

I. Namioka - One of the best experts on this subject based on the ideXlab platform.

  • Kakutani-type fixed point theorems: A survey
    Journal of Fixed Point Theory and Applications, 2011
    Co-Authors: I. Namioka
    Abstract:

    A Kakutani-type fixed point theorem refers to a theorem of the following kind: Given a group or Semigroup S of continuous affine transformations s : Q → Q , where Q is a nonempty compact convex subset of a Hausdorff locally convex linear topological space, then under suitable conditions S has a common fixed point in Q , i.e., a point $${a \in Q}$$ such that s ( a ) = a for each $${s \in S}$$ . In 1938, Kakutani gave two conditions under each of which a common fixed point of S in Q exists. They are (1) the condition that S be a Commutative Semigroup, and (2) the condition that S be an equicontinuous group. The present survey discusses subsequent generalizations of Kakutani’s two theorems above.

B. M. Vernikov - One of the best experts on this subject based on the ideXlab platform.