Vol. 58, No. 2, 1975

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The spectrum of an equational class of groupoids

Jon Froemke and Robert Willis Quackenbush

Vol. 58 (1975), No. 2, 381–386

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
Received: 26 March 1974
Published: 1 June 1975
Jon Froemke
Robert Willis Quackenbush