1,653 research outputs found
Binomial Ideals and Congruences on Nn
ProducciĆ³n CientĆficaA congruence on Nn is an equivalence relation on Nn that is compatible with the additive structure. If k is a field, and I is a binomial ideal in k[X1,ā¦,Xn] (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on Nn by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of Xu and Xv that belongs to I. While every congruence on Nn arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on Nn are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly (Kahle and Miller Algebra Number Theory 8(6):1297ā1364, 2014) with an eye on Eisenbud and Sturmfels (Duke Math J 84(1):1ā45, 1996) and Ojeda and Piedra SĆ”nchez (J Symbolic Comput 30(4):383ā400, 2000).National Science Foundation (grant DMS-1500832)Ministerio de EconomĆa, Industria y Competitividad (project MTM2015-65764-C3-1)Junta de Extremadura (grupo de investigaciĆ³n FQM-024
Growth of generating sets for direct powers of classical algebraic structures
For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),ā¦), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structuresāa group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.Publisher PDFPeer reviewe
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