Vol. 26, No. 3, 1968

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Multiplicity type and subalgebra structure in universal algebras

Matthew Gould

Vol. 26 (1968), No. 3, 469–485

By a universal algebra, or briefly, an algebra we shall here mean a pair A;Fconsisting of a nonvoid set A and a nonvoid set F of finitary operations on A. The multiplicity type of A;F, is the sequence μ = μ01,n,where μn is the cardinality of {f Ff is n-ary}. The class of all algebras of multiplicity type μ is denoted K(μ).

We shall study the relationship between the multiplicity type of an algebra and its family of subalgebras. To this end, we set S(A;F) = {BϕB A and B;F is a subalgebra of A;F⟩} and, for every multiplicity type μ, T(μ) = {S(A;F)A;F⟩∈ K(μ)}. We define a quasi-ordering and an equivalence on the class of multiplicity types as follows. If μ and μare multiplicity types, define μ μif T(μ) T(μ) and μ μif T(μ) = T(μ). We shall give necessary and sufficient conditions for μ μ, in terms of properties of cardinal numbers, and we shall also find a “normal form” for multiplicity types, whereby every multiplicity type will have a unique representation in normal form and the ordering of multiplicity types in normal form will be characterized by relatively simple criteria.

Mathematical Subject Classification
Primary: 08.30
Received: 12 June 1967
Published: 1 September 1968
Matthew Gould