A necessary condition to test the minimality of generalized linear sequential machines using the theory of near-semirings

In this work first we study the properties of nearsemirings to introduce an α-radical. Then we observe the role of near-semirings in generalized linear sequential machines, and we test the minimality through the radical. Introduction Holcombe used the theory of near-rings to study linear sequential machines of Eilenberg [1, 4]. Though the picture of near-rings in linear sequential machines is a natural extension of the syntactic semigroups, the decomposition of linear sequential machines, which is different from the one given by Eilenberg, enabled Holcombe to study these machines thoroughly using near-rings [4]. Holcombe established several properties of machines using near-rings. Indeed, he has introduced an α-radical of affine near-rings which plays an important role to test the minimality of linear sequential machines [5]. The construction of the radical is motivated by Theorem 4.6 of [4], which can be read as: Let M = (Q,A,B, F,G) be a linear sequential machine. If M is minimal then there is no proper nonzero N -submodule K of Q such that This work has been presented in the “69th Workshop on General Algebra", and “20th Conference for Young Algebraists" in Potsdam, March 18-20, 2005 with the title Near-Semirings in Generalized Linear Sequential Machines 2000 Mathematics Subject Classification: 16Y99, 68Q70, 20M11, 20M35.