On the Two-Variable Fragment of the Equational Theory of the Max-Sum Algebra of the Natural Numbers

Abstract

<p>This paper shows that the collection of identities in two variables<br />which hold in the algebra N of the natural numbers with constant<br />zero, and binary operations of sum and maximum does not have a<br />finite equational axiomatization. This gives an alternative proof of the<br />non-existence of a finite basis for N - a result previously obtained by<br />the authors. As an application of the main theorem, it is shown that<br />the language of Basic Process Algebra (over a singleton set of actions),<br />with or without the empty process, has no finite omega-complete equational<br />axiomatization modulo trace equivalence.</p><p><br />AMS Subject Classification (1991): 08A70, 08B05, 03C05, 68Q70.<br />ACM Computing Classification System (1998): F.4.1.<br />Keywords and Phrases: Equational logic, varieties, complete axiomatizations,<br />process algebra, trace equivalence.</p>