The spectrum of an equational class of groupoids

Jon Froemke and Robert Willis Quackenbush

Vol. 58 (1975), No. 2, 381–386

DOI: 10.2140/pjm.1975.58.381

Abstract

The spectrum of an equational class 𝒦 is the set of positive integers Spec(𝒦) = {n|∃A ∈𝒦,|A| = n}. It is obvious that 1 ∈ Spec(𝒦) and x,y ∈ Spec(𝒦) implies xy ∈ Spec(𝒦) for any equational class 𝒦; i.e. Spec(𝒦) is a multiplicative monoid of positive integers. Conversely, G. Grätzer showed that given any multiplicative monoid of positive integers 𝒮 there is an equational class 𝒦 such that 𝒮 = Spec(𝒦). In this paper we show that 𝒦 can be chosen to be an equational class of groupoids.

Mathematical Subject Classification

Primary: 08A15

Secondary: 20L99

Milestones

Received: 26 March 1974

Published: 1 June 1975