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The Algebra Library (alglib)

Collection of algebraic operations.

nullary_relation

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nullary_relation()

true if is a relation of 0-arity.

Note: In case of is a operator, use space after as temporary workaround. eg., nullary_relation(= )

unary_relation

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unary_relation()

true if is a unary relation.

Note: In case of is a operator, use space after as temporary workaround. eg., unary_relation(= )

binary_relation

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binary_relation()

true if is a binary relation.

Note: In case of is a operator, use space after as temporary workaround. eg., binary_relation(= )

ternary_relation

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ternary_relation()

true if is a ternary relation.

Note: In case of is a operator, use space after as temporary workaround. eg., ternary_relation(= )

irreflexive

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irreflexive(D, )

A binary relation is irreflexive if a ∼ a is never true for all a in D.

reflexive

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reflexive(D, )

A binary relation is reflexive if a ∼ a is always true for all a in D.

symmetric

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symmetric(D, )

A binary relation is symmetric if for all a and b in D, whenever a ∼ b is true, then also b ∼ a is also true.

antisymmetric

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antisymmetric(D, )

A binary relation is antisymmetric if for all a and b in D, whenever a ∼ b and b ∼ a are true, then b = a.

transitive

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transitive(D, )

A binary relation is transitive if for all a, b and c in D, whenever a ∼ b and b ∼ c are true, then also a ∼ c is true.

equivalence_relation

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equivalence_relation(D, )

A binary relation is an equivalence_relation if it is reflexive, symmetric, and transitive

unary_relation_substitution_laws

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unary_relation_substitution_laws(D, R, )

A unary relation obeys substitution laws if the relation R produces the same results when values are substituted according to the binary relation .

For example, unary_relation_substitution_laws({1; 2}, {1; 2}, (1, 2)) is true because the result of R(1) remains true when 1 is replaced by 2.

binary_relation_substitution_laws

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binary_relation_substitution_laws(D, R, )

A binary relation obeys substitution laws if the relation R produces the same results when values are substituted according to the binary relation .

preorder

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preorder(D, )

A binary relation is a preorder if it is reflexive and transitive.

partial_order

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partial_order(D, )

A binary relation is a partial order if it is reflexive, antisymmetric, and transitive.

strict_partial_order

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strict_partial_order(D, )

A binary relation is a strict partial order if it is irreflexive, antisymmetric, and transitive. This is similar to a partial order, but is irreflexive rather than reflexive.

comparable

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comparable(a, b, )

Two elements a and b are comparable if either a ≼ b or b ≼ a.

total_order

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total_order(D, )

A binary relation is a total order if it is a partial order, and every element in D is comparable with every other element in D with respect to the partial order.

strict_total_order

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strict_total_order(D, )

A binary relation is a strict total order if it is a strict partial order, and every element in D is either comparable or equal to another element in D.

maximal_element

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maximal_element[D, ]

The maximal element of a partial order is an element which is greater than all other elements in D.

has_maximal_element

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has_maximal_element(D, )

true if the partial order contains a maximal element.

minimal_element

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minimal_element[D, ]

The minimal element of a partial order is an element which is less than all other elements in D.

has_minimal_element

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has_minimal_element(D, )

true if the partial order contains a minimal element.

unary_operator

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unary_operator(D, )

A unary operator takes one argument and returns another argument of the same type, i.e., it can be represented as a binary relation.

binary_operator

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binary_operator(D, )

A binary operator takes two arguments and returns another argument of the same type, i.e., it can be represented as a relation of arity 3.

approximately_commutative

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approximately_commutative(D, , )

A binary operator is approximately commutative if a ⊙ b is approximately equal to b ⊙ a for all a and b in D, where approximately equal is defined by the binary relation .

commutative

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commutative(D, )

A binary operator is commutative if a ⊙ b is equal to b ⊙ a for all a and b in D.

approximately_associative

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approximately_associative(D, , )

A binary operator is approximately associative if (a ⊙ b) ⊙ c is approximately equal to a ⊙ (b ⊙ c) for all a, b and c in D, where approximately equal is defined by the binary relation .

associative

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associative(D, )

A binary operator is associative if (a ⊙ b) ⊙ c is equal to a ⊙ (b ⊙ c) for all a, b and c in D.

idempotent

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idempotent(D, , )

A binary operator is idempotent if a ⊙ a = a for all a in D.

left_identity

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left_identity[D, ]

A left identity of a binary operator is an element i such that i ⊙ a = a for all a in D.

has_left_identity

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has_left_identity(D, )

true if has a left identity.

right_identity

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right_identity[D, ]

A right identity of a binary operator is an element i such that a ⊙ i = a for all a in D.

has_right_identity

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has_right_identity(D, )

true if has a right identity.

identity

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identity[D, ]

An identity of a binary operator is both a left and right identity.

has_identity

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has_identity(D, )

true if has an identity.

approximately_has_inverse

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approximately_has_inverse(D, , N, )

For a binary relation , an inverse of an element a is an element b such that a ⊙ b is approximately equal to the identity element. For approximately_has_inverse, the inverse operator is given as N, so the inverse of a is N[a], and approximately equal to is defined by .

left_zero

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left_zero[D, ]

A left zero of a binary operator is an element z such that z ⊙ a = z for all a in D.

has_left_zero

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has_left_zero(D, )

true if has a left zero.

right_zero

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right_zero[D, ]

A right zero of a binary operator is an element z such that z ⊙ a = z for all a in D.

has_right_zero

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has_right_zero(D, )

true if has a right zero.

approximately_left_distributive

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approximately_left_distributive(D, , , )

A binary operator is approximately left distributive over another binary operator if a ⊗ (b ⊕ c) is approximately equal to (a ⊗ b) ⊕ (a ⊗ c) for all a, b and c in D.

left_distributive

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left_distributive(D, , )

A binary operator is left distributive over another binary operator if a ⊗ (b ⊕ c) is equal to (a ⊗ b) ⊕ (a ⊗ c) for all a, b and c in D.

approximately_right_distributive

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approximately_right_distributive(D, , , )

A binary operator is approximately right distributive over another binary operator if (a ⊕ b) ⊗ c is approximately equal to (a ⊗ c) ⊕ (b ⊗ c) for all a, b and c in D.

right_distributive

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right_distributive(D, , , )

A binary operator is right distributive over another binary operator if (a ⊕ b) ⊗ c is equal to (a ⊗ c) ⊕ (b ⊗ c) for all a, b and c in D.

approximately_distributive

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approximately_distributive(D, , , )

A binary operator is approximately distributive over another binary operator if it is approximately both left and right distributive.

distributive

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distributive(D, , )

A binary operator is distributive over another binary operator if it is both left and right distributive.

absorption_laws

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absorption_laws(D, , )

Binary operator and obey absorption laws if a ⊔ (a ⊓ b) = a = a ⊓ (a ⊔ b) for all a and b in D. The absorption laws are one of the defining characteristics of the meet and join operations of a lattice.

approximately_zero_annihilation

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approximately_zero_annihilation(D, , )

A binary relation obeys approximate zero annihilation if there exists an element that is approximately both a left and right zero.

zero_of_operator

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zero_of_operator[D, ]

A zero of a binary operator is an element that is both a left and right zero.

zero_annihilation

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zero_annihilation(D, )

A binary relation obeys zero annihilation if there exists an element that is both a left and right zero.

semilattice

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semilattice(D, )

A semilattice is a set D, with an operator , such that is commutative, associative, and idempotent. The classic example of a semilattice is a set of subsets with the intersection operator.

bounded_semilattice

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bounded_semilattice(D, )

A bounded semilattice is a semilattice with a maximal element. Equivalently, a semilattice is bounded if it is an idempotent commutative monoid.

lattice

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lattice(D, , )

A lattice is a set D with two operators, and , usually denoted join and meet respectively, such that (D, ⊓) and (D, ⊔) are semilattices, and the operators and obey absorption laws.

meet_bounded_lattice

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meet_bounded_lattice(D, , )

A lattice is meet bounded if the meet operator forms a bounded semilattice.

join_bounded_lattice

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join_bounded_lattice(D, , )

A lattice is join bounded if the join operator forms a bounded semilattice.

bounded_lattice

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bounded_lattice(D, , )

A lattice is bounded if both the meet and join operators form bounded semilattices.

partial_order_and_lattice

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partial_order_and_lattice(D, , , )

A lattice is partially ordered if it has an operator defining a partial order, such that a ≼ b if and only if (a ⊓ b) = a, and (a ⊔ b) = b.

partial_order_and_meet_bounded_lattice

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partial_order_and_meet_bounded_lattice(D, , , )

A partially ordered lattice is meet bounded if the meet operator forms a bounded semilattice, and the identity of is the maximal element of .

partial_order_and_join_bounded_lattice

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partial_order_and_join_bounded_lattice(D, , , )

A partially ordered lattice is join bounded if the join operator forms a bounded semilattice, and the identity of is the minimal element of .

partial_order_and_bounded_lattice

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partial_order_and_bounded_lattice(D, , , )

A partially ordered lattice is bounded if is both meet and join bounded.

approximate_monoid

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approximate_monoid(D, , )

An approximate monoid is a set D with a binary operator that is approximately associative and has an identity element.

monoid

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monoid(D, )

A monoid is a set D with a binary operator that is associative and has an identity element.

approximate_commutative_monoid

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approximate_commutative_monoid(D, , )

An approximate commutative monoid is an approximate monoid where the operator is approximately commutative.

commutative_monoid

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commutative_monoid(D, )

A commutative monoid is a monoid where the operator is commutative.

approximate_group

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approximate_group(D, , N, )

An approximate group is an approximate monoid where the operator has approximate inverses defined by N.

group

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group(D, , N)

A group is a monoid where the operator has inverses defined by N.

approximate_abelian_group

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approximate_abelian_group(D, , N, )

An approximate abelian group is an approximate commutative monoid where the operator has approximate inverses defined by N.

abelian_group

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abelian_group(D, , N)

An abelian group is a commutative monoid where the operator has inverses defined by N.

approximate_semiring

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approximate_semiring(D, , , )

An approximate semiring is a set D with two operators, and , where forms an approximate commutative monoid, forms an approximate monoid and has an approximate zero, distributes over , and the identity of is approximately equal to the zero of .

semiring

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semiring(D, , )

A semiring is an approximate semiring with equality defined as =.

approximate_ring

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approximate_ring(D, , N, , )

An approximate ring is an approximate semiring where the operator has inverses defined by N.

ring

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ring(D, , N, )

A ring is a semiring where the operator has inverses.

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