##### Original version

International Journal of Mathematics. 2019, 30 (02):1950013, DOI: https://doi.org/10.1142/S0129167X19500137

##### Abstract

We introduce and analyze the full [Formula: see text] and the reduced [Formula: see text] Nica–Toeplitz algebra associated to an ideal [Formula: see text] in a right-tensor [Formula: see text]-precategory [Formula: see text] over a right LCM semigroup [Formula: see text]. These [Formula: see text]-algebras unify cross-sectional [Formula: see text]-algebras associated to Fell bundles over discrete groups and Nica–Toeplitz [Formula: see text]-algebras associated to product systems. They also allow a study of Doplicher–Roberts versions of the latter. A new phenomenon is that when [Formula: see text] is not right cancellative then the canonical conditional expectation takes values outside the ambient algebra. Our main result is a uniqueness theorem that gives sufficient conditions for a representation of [Formula: see text] to generate a [Formula: see text]-algebra naturally lying between [Formula: see text] and [Formula: see text]. We also characterize the situation when [Formula: see text]. Unlike previous results for quasi-lattice monoids, [Formula: see text] is allowed to contain nontrivial invertible elements, and we accommodate this by identifying an assumption of aperiodicity of an action of the group of invertible elements in [Formula: see text]. One prominent condition for uniqueness is a geometric condition of Coburn’s type, exploited in the work of Fowler, Laca and Raeburn. Here we shed new light on the role of this condition by relating it to a [Formula: see text]-algebra associated to [Formula: see text] itself.