The Algebra Library (alglib)

nullary_relation

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

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

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

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

irreflexive(D, ∼)

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

reflexive

reflexive(D, ∼)

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

symmetric

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

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

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

equivalence_relation(D, ∼)

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

unary_relation_substitution_laws

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

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

preorder(D, ≼)

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

partial_order

partial_order(D, ≼)

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

strict_partial_order

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

comparable(a, b, ≼)

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

total_order

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

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

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

has_maximal_element(D, ≼)

true if the partial order ≼ contains a maximal element.

minimal_element

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

has_minimal_element(D, ≼)

true if the partial order ≼ contains a minimal element.

unary_operator

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

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

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

commutative(D, ⊙)

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

approximately_associative

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

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

idempotent(D, ⊙, ≈)

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

left_identity

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

has_left_identity(D, ⊙)

true if ⊙ has a left identity.

right_identity

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

has_right_identity(D, ⊙)

true if ⊙ has a right identity.

identity

identity[D, ⊙]

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

has_identity

has_identity(D, ⊙)

true if ⊙ has an identity.

approximately_has_inverse

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

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

has_left_zero(D, ⊙)

true if ⊙ has a left zero.

right_zero

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

has_right_zero(D, ⊙)

true if ⊙ has a right zero.

approximately_left_distributive

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

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

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

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

approximately_distributive(D, ⊗, ⊕, ≈)

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

distributive

distributive(D, ⊗, ⊕)

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

absorption_laws

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

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

zero_of_operator[D, ⊙]

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

zero_annihilation

zero_annihilation(D, ⊙)

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

semilattice

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

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

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

meet_bounded_lattice(D, ⊓, ⊔)

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

join_bounded_lattice

join_bounded_lattice(D, ⊓, ⊔)

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

bounded_lattice

bounded_lattice(D, ⊓, ⊔)

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

partial_order_and_lattice

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

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

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

partial_order_and_bounded_lattice(D, ≼, ⊓, ⊔)

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

approximate_monoid

approximate_monoid(D, ⊙, ≈)

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

monoid

monoid(D, ⊙)

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

approximate_commutative_monoid

approximate_commutative_monoid(D, ⊙, ≈)

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

commutative_monoid

commutative_monoid(D, ⊙)

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

approximate_group

approximate_group(D, ⊙, N, ≈)

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

group

group(D, ⊙, N)

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

approximate_abelian_group

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

abelian_group(D, ⊙, N)

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

approximate_semiring

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

semiring(D, ⊕, ⊙)

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

approximate_ring

approximate_ring(D, ⊕, N, ⊙, ≈)

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

ring

ring(D, ⊕, N, ⊙)

A ring is a semiring where the ⊕ operator has inverses.