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By a universal algebra, or
briefly, an algebra we shall here mean a pair ⟨A;F⟩ consisting 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 = ⟨μ0,μ1,⋯,μn,⋯⟩ where μn is the cardinality of
{f ∈ F∣f 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) = {B∣ϕ≠B ⊆ 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 μ′ are
multiplicity types, define μ ≦ μ′ if T(μ) ⊆ T(μ′) and μ ≡ μ′ if 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.
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