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Special type of lattice
the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice
Distributive_lattice
theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets. Formally, a complete lattice L is
Completely distributive lattice
Completely_distributive_lattice
Set whose pairs have minima and maxima
of lattices can be generalized to semilattices, and some notable subclasses of lattices are Heyting algebras, Boolean algebras, distributive lattices, and
Lattice_(order)
concept of distributivity, applied to the formation of suprema and infima. Most of these apply to partially ordered sets that are at least lattices, but the
Distributivity_(order_theory)
Property involving two mathematical operations
are given in the article distributivity (order theory). This also includes the notion of a completely distributive lattice. In the presence of an ordering
Distributive_property
Glossary of terms used in branch of mathematics
that are not already complete lattices. Completely distributive lattice. A complete lattice is completely distributive if arbitrary joins distribute over
Glossary_of_order_theory
Algebraic structure used in logic
nonempty distributive lattice, in particular every nonempty finite chain, is automatically complete and completely distributive, and hence a Heyting algebra
Heyting_algebra
Existence of certain infima or suprema of a given poset
case the complete lattice X is constructively completely distributive. See also the articles on complete distributivity and distributivity (order theory)
Completeness_(order_theory)
Set with operations obeying given axioms
lattice: a lattice in which arbitrary meet and joins exist. Bounded lattice: a lattice with a greatest element and least element. Distributive lattice: a lattice
Algebraic_structure
Generalized topological space
"logical" structures such as semilattices, distributive lattices, complete and completely distributive lattices, Boolean algebras, complete atomic Boolean
Chu_space
In mathematics, the notions of an absolutely monotonic function and a completely monotonic function are two very closely related concepts. Both imply very
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
(order theory) Dense order Distributivity (order theory) Modular lattice Distributive lattice Completely distributive lattice Ascending chain condition
List_of_order_theory_topics
Nonempty, upper-bounded, downward-closed subset
is no proper filter that is a strict superset. When a poset is a distributive lattice, maximal ideals and filters are necessarily prime, while the converse
Ideal_(order_theory)
Relationship between certain categories
DLat01 of bounded distributive lattices. Hence, DLat01 is dual to CohSp—one obtains Stone's representation theorem for distributive lattices. When restricting
Stone_duality
Partially ordered set equipped with a rank function
fixed N The Boolean lattice of finite subsets of a set ordered by inclusion (number of elements of the subset) Any distributive lattice of finite lower sets
Graded_poset
Mathematical relation inside orderings
covering relation of a Tamari lattice is the skeleton of an associahedron. The covering relation of any finite distributive lattice forms a median graph. On
Covering_relation
Basic operation in mathematical morphology
later being extended to grayscale images, and subsequently to complete lattices. The erosion operation usually uses a structuring element for probing and
Erosion_(morphology)
Vector space equipped with a bilinear product
space is commutative, left distributivity and right distributivity are equivalent, and, in this case, only one distributivity requires a proof. In general
Algebra_over_a_field
Technical treatment of Boolean algebras
lattice basis, it suffices to define a Boolean algebra as a distributive lattice satisfying x∧¬x = 0 and x∨¬x = 1, called a complemented distributive
Boolean algebras canonically defined
Boolean_algebras_canonically_defined
Order-preserving mathematical function
analysis (second ed.). Grätzer, George (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman. ISBN 0-7167-0442-0. Pemberton,
Monotonic_function
Algebraic structure with a binary operation
both left and right semimedial Left distributive If it satisfies the identity x • yz ≡ xy • xz Right distributive If it satisfies the identity yz • x
Magma_(algebra)
Algebraic structure in linear algebra
field F into the endomorphism ring of this group. Specifically, the distributivity of scalar multiplication with respect to vector addition means that
Vector_space
Hungarian and American mathematician and physicist (1903–1957)
Consequently, the distributive law of classical logic must be replaced with a weaker condition. Instead of a distributive lattice, propositions about
John_von_Neumann
Generalization of vector spaces from fields to rings
a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module
Module_(mathematics)
Equalities for combinations of sets
{\displaystyle S,} ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean
List of set identities and relations
List_of_set_identities_and_relations
Certain topology in mathematics
and the given topology on X coincide. The order topology makes X into a completely normal Hausdorff space. The standard topologies on R, Q, Z, and N are
Order_topology
Algebraic manipulation of "true" and "false"
axiomatization of Boolean algebra, such as the axioms for a complemented distributive lattice, a sufficient condition for an algebraic structure of this kind to
Boolean_algebra
Generalization of the concept of subsequence to the case of nets
"subsequence" for nets is the notion of a "subnet". The definition is not completely straightforward, but is designed to allow as many theorems about subsequences
Subnet_(mathematics)
Theories in mathematical logic
convention in set theory given above. The axioms are: The axioms for a distributive lattice (see above) ∀a a∧¬a = 0, ∀a a∨¬a = 1 (properties of negation) Some
List_of_first-order_theories
Alternative mathematical ordering
[ a , x , b ] {\displaystyle [a,x,b]} . The system of open intervals completely defines the cyclic order and can be used as an alternate definition of
Cyclic_order
Abstract mathematics relationship
theorem stating a duality between finite partial orders and finite distributive lattices. In pointless topology the category of spatial locales is known
Equivalence_of_categories
Mathematical ranking of a set
weak orderings is the graph describing the covering relation of the face lattice of the permutohedron. For instance, for n = 3 , {\displaystyle n=3,} the
Weak_ordering
Algebraic structure with addition, multiplication, and division
1/a, called the multiplicative inverse of a, such that a ⋅ a−1 = 1. Distributivity of multiplication over addition: a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c). An
Field_(mathematics)
Hypercomplex number system
quaternions. Multiplication of octonions is more complex. Multiplication is distributive over addition, so the product of two octonions can be calculated by summing
Octonion
Mathematical concept
Birkhoff's representation theorem, an equivalence between finite distributive lattices (the lattice of open sets of the topology) and partial orders (the partial
Finite_topological_space
Number representing a continuous quantity
{\displaystyle (ab)c=a(bc)} for all real numbers a, b and c. Multiplication is distributive over addition, which means that a ( b + c ) = a b + a c {\displaystyle
Real_number
Propositional calculus in which there are more than two truth values
form a complete lattice with an infinite distributive law, which defines a unique complete Heyting algebra structure on the lattice. Implication → L
Many-valued_logic
Type of logical system
Predicate functor logic, primarily by Willard Quine. These algebras are all lattices that properly extend the two-element Boolean algebra. Tarski and Givant
First-order_logic
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COMPLETELY DISTRIBUTIVE-LATTICE
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COMPLETELY DISTRIBUTIVE-LATTICE
COMPLETELY DISTRIBUTIVE-LATTICE
COMPLETELY DISTRIBUTIVE-LATTICE
COMPLETELY DISTRIBUTIVE-LATTICE
COMPLETELY DISTRIBUTIVE-LATTICE
COMPLETELY DISTRIBUTIVE-LATTICE
COMPLETELY DISTRIBUTIVE-LATTICE
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