The main goal of this presentation is to study the algebraic properties of the deterministic processes with dynamic represented by a homogeneous linear recurrence over the field C. We delve into the subsets of C to see if the dynamic of the given process is also a homogeneous linear recurrence over given subset in certain conditions. The challenge appears when we get out from comfort zone, given by field properties. So, we start with an overview of homogeneous linear recurrent processes over C and its subsets. We remind the main definitions and properties from [5], like generating vector, characteristic polynomial and minimality. Also, we formulate the minimization method based on matrix rank definition, which was theoretically grounded in [6]. Next, we go deeper into homogeneous linear recurrent processes over numerical rings. We formulate and prove necessary and sufficient conditions for a homogeneous linear recurrence over C to be also a homogeneous linear recurrence over a subfield or subring, like R, Q, Z or an extension field of Q. After that, we are interested in recurrence criteria over sign-based ring subsets. We split the ring into two subsets, one containing the positive elements and the second containing the negative ones. Based on results from [1] it is shown that the recurrence criteria other these subsets are based on the number of positive real roots of the minimal characteristic polynomial over that ring and, in the most complex case when it is a single one, they are also based on the relationship of that positive real root with the rest of the roots. The last part is dedicated to deterministic processes with dynamic represented by a Littlewood, Newman or Borwein homogeneous linear recurrence. Mainly, these are homogeneous linear recurrences over subsets of f¡1; 0; 1g. Several results are presented, based on the properties of Littlewood, Newman and Borwein polynomials developed in [2-4] and [7].