Package org.uacalc.alg

What is an algebra? and how do we represent it in a computer? We bascially follow Algebras, Lattices, Varieties by McKenzie, McNulty, Taylor.

See: Description

Interface Summary

Interface	Description
Algebra
SmallAlgebra	A small algebra is one whose universe can be effectively indexed by {0,..., n-1} for some positive int n.

Class Summary

Class	Description
AlgebraFromMinimalSets	Starting with an algebra B which is permutational (nonconstant unary polynomials are permutations) and a geometry for the minimal sets, this constructs an algebra A with tame minimal sets, having B as a minimal set.
Algebras	A class with static methods for algebras.
AlgebraWithGeneratingVector
BasicAlgebra	This class represents SmallAlgebra's.
BigProductAlgebra	This class represents the direct product of SmallAlgebras which is too big to be a SmallAlgebra.
Closer	A class for finding the closure with configurations for several options and fields to hold side results.
CloserTiming	A class to hold the data for the timing information in the UI.
FreeAlgebra	This class represents a subalgebra of a direct product of SmallAlgebras.
GeneralAlgebra	This class represents general algebras that may or may not be "computer finite".
Homomorphism	A homomorphism from the domain algebra into the range algebra.
Malcev	A class with static methods for Mal'cev conditions such as finding Jonsson terms, etc.
MaltsevDecompositionIterator	An iterator for idempotent algebras giving sections, this is, quotients of subalgebras of the given algebra.
MaltsevProductDecomposition	Suppose P is a robust property; a property such that if A is an idempotent algebra in the Maltsev product of two idempotent varieties satisfying P, then A satisfies P.
MatrixPowerAlgebra
ParameterizedAlgebra	This class represents SmallAlgebra's.
PolinLikeAlgebra	Given a homomorphism f: A to B, this constructs a Polin type algebra on the disjoint union of A and B.
PowerAlgebra	This class represents the direct product of SmallAlgebras.
ProductAlgebra	This class represents the direct product of SmallAlgebras.
QuotientAlgebra	This class represents a quotient algebra of a SmallAlgebra.
QuotientElement	This class represents an element in a quotient algebra.
ReductAlgebra	This class represents a reduct of a SmallAlgebra to a list of Terms.
Subalgebra	This class represents a subalgebra of a SmallAlgebra.
SubProductAlgebra	This class represents a subalgebra of a direct product of SmallAlgebras.
UnaryTermsMonoid	The monoid or semigroup of unary terms.

Enum Summary

Enum	Description
SmallAlgebra.AlgebraType

Package org.uacalc.alg Description

What is an algebra? and how do we represent it in a computer? We bascially follow Algebras, Lattices, Varieties by McKenzie, McNulty, Taylor. The java interface Algebra describes the methods an algebra should have.

How the algebra is represented internally depends on its size and its origin: did the user input it? was it constructed as a direct product of algebra? Also there needs to be different a representation for certain kinds of algebras such as lattices.

For an algebra of size n, where n is reasonably small, we will have a map between the universe of the algebra and the set {0, 1, ..., n-1} and the operations will be represented by tables on this set of integers. But for large algebras tables may be impractical or impossible. So if A has 7 elements and one binary operation, we might generate the multiplication tables for A² but not for A³ and certainly not for A⁴. Of course for A³ the operations would be computed componentwise.