A Correspondence between Balanced Varieties and Inverse Monoids

There is a well-known correspondence between varieties of algebras and fully invariant congruences on the appropriate term algebra. A special class of varieties are those which are balanced, meaning they can be described by equations in which the same variables appear on each side. In this paper, we prove that the above correspondence, restricted to balanced varieties, leads to a correspondence between balanced varieties and inverse monoids. In the case of unary algebras, we recover the theorem of Meakin and Sapir that establishes a bijection between congruences on the free monoid with n generators and wide, positively self-conjugate inverse submonoids of the polycyclic monoid on n generators. In the case of varieties generated by linear equations, meaning those equations where each variable occurs exactly once on each side, we can replace the clause monoid above by the linear clause monoid. In the case of algebras with a single operation of arity n, we prove that the linear clause mo...

• Type: Journal
• Year: 2006
• Where: IJAC
• Authors: Mark V. Lawson
• Keywords: Algebras, Balanced, IJAC 2006, Monoids