The Algebra Library (alglib)

Collection of algebraic operations.

unary_relation

unary_relation(D, ∼)

Returns true if is a unary relation over domain D.

Definition

@inline
def unary_relation(D, ) =
equal(arity[], 1)

binary_relation

binary_relation(D, ∼)

Returns true if is a binary relation over domain D.

Definition

@inline
def binary_relation(D, ) =
equal(arity[], 2)

irreflexive

irreflexive(D, ∼)

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

Definition

@inline
def irreflexive(D, ) =
binary_relation(D, ) and
forall(a ∈ D: ¬(a a))

reflexive

reflexive(D, ∼)

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

Definition

@inline
def reflexive(D, ) =
binary_relation(D, ) and
forall(a ∈ D: a a)

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.

Definition

@inline
def symmetric(D, ) =
binary_relation(D, ) and
forall(a ∈ D, b ∈ D: a b b a)

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.

Definition

@inline
def antisymmetric(D, ) =
binary_relation(D, ) and
binary_relation_substitution_laws(D, , =) and
forall(a ∈ D, b ∈ D: ((a b) and (b a)) a = b)

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.

Definition

@inline
def transitive(D, ) =
binary_relation(D, ) and
forall(a ∈ D, b ∈ D, c ∈ D: a b and b c implies a c)

equivalence_relation

equivalence_relation(D, ∼)

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

Definition

@inline
def equivalence_relation(D, ) =
reflexive(D, ) and symmetric(D, ) and transitive(D, )

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.

Definition

@inline
def unary_relation_substitution_laws(D, R, ) =
unary_relation(D, R) and binary_relation(D, ) and
forall(a ∈ D, b ∈ D: (a b) (R(a) R(b)))

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 .

Definition

@inline
def binary_relation_substitution_laws(D, , ) =
binary_relation(D, ) and binary_relation(D, ) and
forall(a1 ∈ D, a2 ∈ D: a1 a2 forall(b ∈ D: a1 b a2 b)) and
forall(b1 ∈ D, b2 ∈ D: b1 b2 forall(a ∈ D: a b1 a b2))

partial_order

preorder(D, ≼)

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

Definition

@inline
def partial_order(D, ) =
reflexive(D, ) and transitive(D, )

partial_order

partial_order(D, ≼)

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

Definition

@inline
def partial_order(D, ) =
reflexive(D, ) and antisymmetric(D, ) and transitive(D, )

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.

Definition

@inline
def strict_partial_order(D, ) =
irreflexive(D, ) and antisymmetric(D, ) and transitive(D, )

comparable

comparable(a, b, ≼)

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

Definition

@inline
def comparable(a, b, ) =
(ab) (ba)

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.

Definition

@inline
def total_order(D, ) =
partial_order(D, ) and
forall(a ∈ D, b ∈ D: comparable(a, b, ))

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.

Definition

@inline
def strict_total_order(D, ) =
strict_partial_order(D, ) and
forall(a ∈ D, b ∈ D: comparable(a, b, ) (b = a))

maximal_element

maximal_element[D, ≼]

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

Definition

@inline
def maximal_element[D, ] =
max_elem: D(max_elem) and
partial_order(D, ) and
forall(a ∈ D: amax_elem)

has_maximal_element

has_maximal_element(D, ≼)

This returns true if the partial order contains a maximal element.

Definition

@inline
def has_maximal_element(D, ) =
partial_order(D, ) and
exists(max_elem ∈ D: forall(a ∈ D: amax_elem))

minimal_element

minimal_element[D, ≼]

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

Definition

@inline
def minimal_element[D, ] =
min_elem: D(min_elem) and
partial_order(D, ) and
forall(a ∈ D: min_elema)

has_minimal_element

has_minimal_element(D, ≼)

This returns true if the partial order contains a minimal element.

Definition

@inline
def has_minimal_element(D, ) =
partial_order(D, ) and
exists(min_elem ∈ D: forall(a ∈ D: min_elema))

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.

Definition

@inline
def unary_operator(D, ) =
equal(arity[], 2)

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.

Definition

@inline
def binary_operator(D, ) =
equal(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 .

Definition

@inline
def approximately_commutative(D, , ) =
binary_operator(D, ) and
reflexive(D, ) and
symmetric(D, ) and
forall(a ∈ D, b ∈ D: (a b) (b a))

commutative

commutative(D, ⊙)

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

Definition

@inline
def commutative(D, ) =
approximately_commutative(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 .

Definition

@inline
def approximately_associative(D, , ) =
binary_operator(D, ) and
reflexive(D, ) and
symmetric(D, ) and
forall(a ∈ D, b ∈ D, c ∈ D: ((a b) c) (a (b c)))

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.

Definition

@inline
def associative(D, ) =
approximately_associative(D, , =)

idempotent

idempotent(D, ⊙, ≈)

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

Definition

@inline
def idempotent(D, ) =
binary_operator(D, ) and
forall(a ∈ D: (a a) = a)

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.

Definition

@inline
def left_identity[D, ] =
ident: D(ident) and
forall(a ∈ D: ((ident a) = a)) and
binary_operator(D, )

has_left_identity

has_left_identity(D, ⊙)

This returns true if has a left identity.

Definition

@inline
def has_left_identity(D, ) =
binary_operator(D, ) and
equivalence_relation(D, =) and
// TODO: ideally we could do
// exists(ident ∈ D: left_identity[D, ⊙](ident))
// however this doesn't terminate
exists(ident ∈ D: forall(a ∈ D: ((ident a) = a)))

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.

Definition

@inline
def right_identity[D, ] =
ident: D(ident) and
forall(a ∈ D: ((a ident) = a)) and
binary_operator(D, )

has_right_identity

has_right_identity(D, ⊙)

This returns true if has a right identity.

Definition

@inline
def has_right_identity(D, ) =
binary_operator(D, ) and
exists(ident ∈ D: forall(a ∈ D: ((a ident) = a)))

identity

identity[D, ⊙]

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

Definition

@inline
def identity[D, ] =
ident: D(ident) and
left_identity[D, ](ident) and right_identity[D, ](ident)

has_identity

has_identity(D, ⊙)

This returns true if has an identity.

Definition

@inline
def has_identity(D, ) =
binary_operator(D, ) and
exists(ident ∈ D: forall(a ∈ D: (a ident) = a and (ident a) = a))

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 .

Definition

@inline
def approximately_has_inverse(D, , N, ) =
binary_operator(D, ) and
has_identity(D, ) and
unary_operator(D, N) and
reflexive(D, ) and
symmetric(D, ) and
forall(a, i: D(a) and identity[D, ](i) implies ((a N[a]) i) and ((N[a] a) i))

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.

Definition

@inline
def left_zero[D, ] =
zero: D(zero) and
forall(a ∈ D: ((zero a) = zero)) and
binary_operator(D, )

has_left_zero

has_left_zero(D, ⊙)

This returns true if has a left zero.

Definition

@inline
def has_left_zero(D, ) =
binary_operator(D, ) and
exists(zero ∈ D: forall(a ∈ D: ((zero a) = 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.

Definition

@inline
def right_zero[D, ] =
zero: D(zero) and
forall(a ∈ D: ((a zero) = zero)) and
binary_operator(D, )

has_right_zero

has_right_zero(D, ⊙)

This returns true if has a right zero.

Definition

@inline
def has_right_zero(D, ) =
binary_operator(D, ) and
exists(zero ∈ D: forall(a ∈ D: ((a zero) = 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.

Definition

@inline
def approximately_left_distributive(D, , , ) =
binary_operator(D, ) and
binary_operator(D, ) and
reflexive(D, ) and
symmetric(D, ) and
forall(a ∈ D, b ∈ D, c ∈ D: (a (b c)) ((a b) (a c)))

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.

Definition

@inline
def left_distributive(D, , ) =
approximately_left_distributive(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.

Definition

@inline
def approximately_right_distributive(D, , , ) =
binary_operator(D, ) and
binary_operator(D, ) and
reflexive(D, ) and
symmetric(D, ) and
forall(a ∈ D, b ∈ D, c ∈ D: ((a b) c) ((a c) (b c)))

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.

Definition

@inline
def right_distributive(D, , ) =
approximately_right_distributive(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.

Definition

@inline
def approximately_distributive(D, , , ) =
approximately_left_distributive(D, , , ) and
approximately_right_distributive(D, , , )

distributive

distributive(D, ⊗, ⊕)

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

Definition

@inline
def distributive(D, , ) =
approximately_distributive(D, , , =)

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.

Definition

@inline
def absorption_laws(D, , ) =
binary_operator(D, ) and
binary_operator(D, ) and
forall(a ∈ D, b ∈ D: (a(ab)) = a = (a(ab)))

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.

Definition

@inline
def approximately_zero_annihilation(D, , ) =
binary_operator(D, ) and
reflexive(D, ) and
symmetric(D, ) and
exists(zero ∈ D: forall(a ∈ D: ((a zero) zero) and ((zero a) 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.

Definition

@inline
def zero_of_operator[D, ] =
zero: D(zero) and
left_zero[D, ](zero) and
right_zero[D, ](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.

Definition

@inline
def zero_annihilation(D, ) =
approximately_zero_annihilation(D, , =)

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.

Definition

@inline
def semilattice(D, ) =
commutative(D, ) and
associative(D, ) and
idempotent(D, )

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.

Definition

@inline
def bounded_semilattice(D, ) =
semilattice(D, ) and
commutative_monoid(D, ) and
has_identity(D, )

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.

Definition

@inline
def lattice(D, , ) =
semilattice(D, ) and
semilattice(D, ) and
absorption_laws(D, , )

meet_bounded_lattice

meet_bounded_lattice(D, ⊓, ⊔)

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

Definition

@inline
def meet_bounded_lattice(D, , ) =
lattice(D, , ) and
bounded_semilattice(D, )

join_bounded_lattice

join_bounded_lattice(D, ⊓, ⊔)

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

Definition

@inline
def join_bounded_lattice(D, , ) =
lattice(D, , ) and
bounded_semilattice(D, )

bounded_lattice

bounded_lattice(D, ⊓, ⊔)

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

Definition

@inline
def bounded_lattice(D, , ) =
meet_bounded_lattice(D, , ) and
join_bounded_lattice(D, , )

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.

Definition

@inline
def partial_order_and_lattice(D, , , ) =
partial_order(D, ) and
lattice(D, , ) and
forall(a ∈ D, b ∈ D: (ab) ((ab) = a)) and
forall(a ∈ D, b ∈ D: (ab) ((ab) = 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 .

Definition

@inline
def partial_order_and_meet_bounded_lattice(D, , , ) =
partial_order_and_lattice(D, , , ) and
meet_bounded_lattice(D, , ) and
has_minimal_element(D, ) and
identity[D, ] maximal_element[D, ]

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 .

Definition

@inline
def partial_order_and_join_bounded_lattice(D, , , ) =
partial_order_and_lattice(D, , , ) and
join_bounded_lattice(D, , ) and
has_maximal_element(D, ) and
identity[D, ] minimal_element[D, ]

partial_order_and_bounded_lattice

partial_order_and_bounded_lattice(D, ≼, ⊓, ⊔)

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

Definition

@inline
def partial_order_and_bounded_lattice(D, , , ) =
partial_order_and_meet_bounded_lattice(D, , , ) and
partial_order_and_join_bounded_lattice(D, , , )

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.

Definition

@inline
def approximate_monoid(D, , ) =
approximately_associative(D, , ) and
has_identity(D, )

monoid

monoid(D, ⊙)

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

Definition

@inline
def monoid(D, ) =
approximate_monoid(D, , =) and
associative(D, )

approximate_commutative_monoid

approximate_commutative_monoid(D, ⊙, ≈)

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

Definition

@inline
def approximate_commutative_monoid(D, , ) =
approximate_monoid(D, , ) and
approximately_commutative(D, , )

commutative_monoid

commutative_monoid(D, ⊙)

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

Definition

@inline
def commutative_monoid(D, ) =
approximate_commutative_monoid(D, , =) and
commutative(D, )

approximate_group

approximate_group(D, ⊙, N, ≈)

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

Definition

@inline
def approximate_group(D, , N, ) =
approximate_monoid(D, , ) and
approximately_has_inverse(D, , N, )

group

group(D, ⊙, N)

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

Definition

@inline
def group(D, , N) =
approximate_group(D, , N, =) and
monoid(D, ) and
approximately_has_inverse(D, , 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.

Definition

@inline
def approximate_abelian_group(D, , N, ) =
approximate_group(D, , N, ) and
approximate_commutative_monoid(D, , )

abelian_group

abelian_group(D, ⊙, N)

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

Definition

@inline
def abelian_group(D, , N) =
approximate_abelian_group(D, , N, =) and
group(D, , N) and
commutative_monoid(D, )

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 .

Definition

@inline
def approximate_semiring(D, , , ) =
approximate_commutative_monoid(D, , ) and
approximate_monoid(D, , ) and
approximately_distributive(D, , , ) and
approximately_zero_annihilation(D, , ) and
exists(x, y: zero_of_operator[D, ](x) and identity[D, ](y) implies x y)

semiring

semiring(D, ⊕, ⊙)

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

Definition

@inline
def semiring(D, , ) =
approximate_semiring(D, , , =) and
commutative_monoid(D, ) and
monoid(D, ) and
distributive(D, , ) and
zero_annihilation(D, ) and
zero_of_operator[D, ] identity[D, ]

approximate_ring

approximate_ring(D, ⊕, N, ⊙, ≈)

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

Definition

@inline
def approximate_ring(D, , N, , ) =
approximate_abelian_group(D, , N, ) and
approximate_semiring(D, , , )

ring

ring(D, ⊕, N, ⊙)

A ring is a semiring where the operator has inverses.

Definition

@inline
def ring(D, , N, ) =
approximate_ring(D, , N, , =) and
abelian_group(D, , N) and
semiring(D, , )