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Admissible orders on terms (power-products of finitely many indeterminates X1,..., Xn play a fundamental role in the definition and construction of Groebner bases for polynomial ideals (see [Bul). By passage to exponents, these orders may be construed as linear orders on [EQUATION] compatible with addition and with smallest element [EQUATION] = (0,...,0). Any such order extends uniquely to a linear order < on [EQUATION] turning ([EQUATION], +, >) into an ordered group such that all elements of [EQUATION] are non-negative, Conversely, any restriction of such an order to [EQUATION] is an admissible order on [EQUATION]. So from now on an "admissible order" will be a linear group order on [EQUATION] with [EQUATION] >= 0.