We give a detailed account of algorithms and applications provided with SINGULAR:PLURAL (we call it PLURAL for short). The poster is done in form of (big) one-page introduction to the capabilities of the system.

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In this paper we introduce an algebra embedding ι : K X → S from the free associative algebra K X generated by a finite or countable set X into the skew monoid ring S = P * Σ defined by the commutative polynomial ring P = K[X × N * ] and by the monoid Σ = σ generated by a suitable endomorphism σ : P → P . If P = K[X] is any ring of polynomials in a countable set of commuting variables, we present also a general Gröbner bases theory for graded two-sided ideals of the graded algebra S = i S i with S i = P σ i and σ : P → P an abstract endomorphism satisfying compatibility conditions with ordering and divisibility of the monomials of P . Moreover, using a suitable grading for the algebra P compatible with the action of Σ, we obtain a bijective correspondence, preserving Gröbner bases, between graded Σ-invariant ideals of P and a class of graded two-sided ideals of S. By means of the embedding ι this results in the unification, in the graded case, of the Gröbner bases theories for commutative and non-commutative polynomial rings. Finally, since the ring of ordinary difference polynomials P = K[X × N] fits the proposed theory one obtains that, with respect to a suitable grading, the Gröbner bases of finitely generated graded ordinary difference ideals can be computed also in the operators ring S and in a finite number of steps up to some fixed degree.2000 Mathematics Subject Classification. Primary 16Z05. Secondary 13P10, 68W30.

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