Rel
REFERENCE
Libraries
alglib

# The Algebra Library (alglib)

Collection of algebraic operations.

## 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, ⊙)

## 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 ⊔ (a ⊓ b)) = a = (a ⊓ (a ⊔ b)))

## 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(∼ ) and
binary_relation_substitution_laws(D, ∼, =) and
forall(a ∈ D, b ∈ D: ((a ∼ b) and (b ∼ a)) ⇒ a = b)

## 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, ⊙, ≈)

## 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, ⊙, ≈)

## 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, ≈)

## 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, ⊙)

## 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, ⊕, ⊙, ≈)

## 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)

## 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)))

## 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))

## 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, ⊗, ⊕, ≈)

## 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))

## 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)))

## 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)))

## 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)))

## 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, ⊙, =)

## 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)

## 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(= )

Definition

@inline
def binary_relation( ∼ ) = equal(arity[∼], 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 ≈.

Definition

@inline
def binary_relation_substitution_laws(D, ∼, ≈) =
binary_relation(∼ ) and binary_relation(≈ ) 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))

## 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, ⊓, ⊔)

## 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, ⊓)

## 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, ⊙, =)

## 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, ⊙)

## comparable#

comparable(a, b, ≼)

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

Definition

@inline
def comparable(a, b, ≼) =
(a ≼ b) ∨ (b ≼ a)

## 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, ⊗, ⊕, =)

## 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, ∼)

## 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, =)

## has_identity#

has_identity(D, ⊙)

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))

## has_left_identity#

has_left_identity(D, ⊙)

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)))

## has_left_zero#

has_left_zero(D, ⊙)

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)))

## has_maximal_element#

has_maximal_element(D, ≼)

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: a ≼ max_elem))

## has_minimal_element#

has_minimal_element(D, ≼)

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_elem ≼ a))

## has_right_identity#

has_right_identity(D, ⊙)

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)))

## has_right_zero#

has_right_zero(D, ⊙)

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)))

## 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)

## 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)

## 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(∼ ) and
forall(a ∈ D: ¬(a ∼ a))

## 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, ⊔)

## 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, ⊓, ⊔)

## 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, ⊗, ⊕, =)

## 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, ⊙)

## 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, ⊙)

## 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: a ≼ max_elem)

## 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, ⊓)

## 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_elem ≼ a)

## 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, ⊙)

## 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(= )

Definition

@inline
def nullary_relation( ∼ ) = equal(arity[∼], 0)

## 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, ≼)

## 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, ≼, ⊓, ⊔)

## 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
equal(identity[D, ⊔], minimal_element[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: (a ≼ b) ≡ ((a ⊓ b) = a)) and
forall(a ∈ D, b ∈ D: (a ≼ b) ≡ ((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 ≼.

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
equal(identity[D, ⊓], maximal_element[D, ≼])

## preorder#

preorder(D, ≼)

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

Definition

@inline
def preorder(D, ≼) =
reflexive(D, ≼) and transitive(D, ≼)

## 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(∼ ) and
forall(a ∈ D: a ∼ a)

## 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, ⊗, ⊕, =)

## 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, ⊙)

## 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, ⊙)

## 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, ⊕, ⊙)

## 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, ⊓)

## 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
equal(zero_of_operator[D, ⊙], identity[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, ≺)

## 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))

## 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(∼ ) and
forall(a ∈ D, b ∈ D: a ∼ b ≡ b ∼ a)

## 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(= )

Definition

@inline
def ternary_relation( ∼ ) = equal(arity[∼], 3)

## 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, ≼))

## 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(∼ ) and
forall(a ∈ D, b ∈ D, c ∈ D: a ∼ b and b ∼ c implies a ∼ c)

## 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)

## 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(= )

Definition

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

## 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(R) and binary_relation(≈ ) and
forall(a ∈ D, b ∈ D: (a ≈ b) ⇒ (R(a) ≡ R(b)))

## 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, ⊙, =)

## 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)