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Growth in varieties of multioperator algebras and Groebner bases in operads. (arXiv:1705.03356v2 [math.RA] UPDATED)

来源于:arXiv
We discuss algorithmic approach to growth of the codimension sequences of varieties of multilinear algebras, or, equivalently, the sequences of the component dimensions of algebraic operads. The (exponentional) generating functions of such sequences are called codimension series of varieties, or generating series of operads. We show that in general there does not exist an algorithm to decide whether the growth exponent of a codimension sequence of a variety defined by given finite sets of operations and identities is equal to a given rational number. In particular, we solve negatively a recent conjecture by Bremner and Dotsenko by showing that the set generating series of binary quadratic operads with bounded number of generators is infinite. Then we recall algorithms which in many cases calculate the codimension series in the form of a defining algebraic or differential equation. For a more general class of varieties, these algorithms give upper and lower bounds for the codimensions i 查看全文>>