“Minimal algebra”的意思、由来-开放百科全书

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Minimal algebra

释义

Definition
Classification
References

Minimal algebra is an important concept in tame congruence theory, a theory that has been developed by Ralph McKenzie and David Hobby ^[1].

Definition

A minimal algebra is a finite algebra with more than one element, in which every non-constant unary polynomial is a permutation on its domain.

Classification

A polynomial of an algebra is a composition of its basic operations, -ary operations and the projections. Two algebras are called polynomially equivalent if they have the same universe and precisely the same polynomial operations. A minimal algebra falls into one of the following types (P. P. Pálfy) ^[1] ^[2]

is of type , or unary type, iff , where denotes the universe of , denotes the set of all polynomials of an algebra and is a subgroup of the symmetric group over .

is of type , or affine type, iff is polynomially equivalent to a vector space.

is of type , or Boolean type, iff is polynomially equivalent to a two-element Boolean algebra.

is of type , or lattice type, iff is polynomially equivalent to a two-element lattice.

is of type , or semilattice type, iff is polynomially equivalent to a two-element semilattice.

References

1. ^¹{{cite book |last1=Hobby |first1=David |last2=McKenzie |first2=Ralph |title=The structure of finite algebras |date=1988 |publisher=American Mathematical Society |location=Providence, RI |isbn=0-8218-5073-3 |page=xii+203 pp}}
2. ^{{cite journal |last1=Pálfy |first1=P. P. |title=Unary polynomials in algebras. I |journal=Algebra Universalis |date=1984 |volume=18 |issue=3 |pages=262-273}}

1 : Algebra

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