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Partial order with joins
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset
Semilattice
Set whose pairs have minima and maxima
theory and universal algebra. The class of lattices can be generalized to semilattices, and some notable subclasses of lattices are Heyting algebras, Boolean
Lattice_(order)
Partition of space by hyperplanes
arrangement of planes. The intersection semilattice L(A) is a meet semilattice and more specifically is a geometric semilattice. If the arrangement is linear or
Arrangement_of_hyperplanes
Algebraic structure
theorem for commutative semigroups in terms of semilattices. A semilattice (or more precisely a meet-semilattice) (L, ≤) is a partially ordered set where every
Semigroup
Concept in order theory
pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that
Join_and_meet
Partially ordered set in which all subsets have both a supremum and infimum
called a closed sublattice of L. The terms complete meet-semilattice or complete join-semilattice is another way to refer to complete lattices since arbitrary
Complete_lattice
least lattices, but the concept can in fact reasonably be generalized to semilattices as well. Probably the most common type of distributivity is the one defined
Distributivity_(order_theory)
Mathematical ordering with upper bounds
(contrast partially ordered sets, which need not be directed). Join-semilattices (which are partially ordered sets) are directed sets as well, but not
Directed_set
In abstract algebra, a branch of mathematics, a maximal semilattice quotient is a commutative monoid derived from another commutative monoid by making
Maximal_semilattice_quotient
Algebraic structure modeling logical operations
generalized Boolean algebra, while (B, ∨, 0) is a generalized Boolean semilattice. Generalized Boolean lattices are exactly the ideals of Boolean lattices
Boolean_algebra_(structure)
Structure in group theory (in mathematics)
semigroup) and idempotents commute (that is, the idempotents of S form a semilattice). Every L {\displaystyle {\mathcal {L}}} -class and every R {\displaystyle
Inverse_semigroup
Mathematical concept
valuable alternative presentation. In the case of semilattices, an explicit construction of the free semilattice F ∨ ( X ) {\displaystyle F_{\vee }(X)} is straightforward
Free_lattice
Type of data structure
should compute the join for any pair of replica states, and should form a semilattice with the initial state as the neutral element. In particular, this means
Conflict-free replicated data type
Conflict-free_replicated_data_type
In mathematics, an algebraic structure
residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras
Residuated_lattice
Important problem in lattice theory
following semilattice-theoretical formulation of CLP. Semilattice-theoretical formulation of CLP: Is every distributive (∨,0)-semilattice isomorphic
Congruence_lattice_problem
sup I then c is an element of I. If the poset P additionally is a join-semilattice (i.e., if it has binary suprema) then these conditions are equivalent
Compact_element
y ≤ x. A model of the Plotkin powertheory is a continuous semilattice: it is a semilattice whose carrier is a domain and for which the operation is continuous
Power_domains
order theory, a nucleus is a function F {\displaystyle F} on a meet-semilattice A {\displaystyle {\mathfrak {A}}} such that (for every p {\displaystyle
Nucleus_(order_theory)
Mathematical set with an ordering
Scott continuity – continuity of a function between two partial orders. Semilattice – Partial order with joins Semiorder – Numerical ordering with a margin
Partially_ordered_set
Generalized alphabetical order
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Lexicographic_order
Well-quasi-ordering of finite trees
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Kruskal's_tree_theorem
Set with operations obeying given axioms
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Algebraic_structure
Glossary of terms used in branch of mathematics
relation. Synonym for Connected relation. Complete semilattice. The notion of a complete semilattice is defined in different ways. As explained in the
Glossary_of_order_theory
Semigroup in which every element is idempotent
independently in the early 1970s by Biryukov, Fennemore and Gerhard. Semilattices, left-zero bands, right-zero bands, rectangular bands, normal bands,
Band_(algebra)
action algebra is an algebraic structure which is both a residuated semilattice and a Kleene algebra. It adds the star or reflexive transitive closure
Action_algebra
Order whose elements are all comparable
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗
Total_order
the inverse semigroup of isomorphisms between principal ideals of a semilattice (a commutative semigroup of idempotents). Munn semigroups are named for
Munn_semigroup
Algebraic structure with addition and multiplication
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Ring_(mathematics)
Vector space equipped with a bilinear product
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Algebra_over_a_field
Relationship between elements of two sets
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗
Binary_relation
complemented lattice, i.e. a Boolean algebra. The same holds for any semilattice when "semilattice" is substituted for "distributive lattice" and "subsemilattice"
Subdirect_product
Concept in the mathematics of partial orders
cannot be a lattice (or even a meet semilattice), since by definition, every two elements in a lattice (or meet semilattice) must have a common lower bound
Strong_antichain
Topic in abstract algebra
commutative, therefore semilattices (one of them is the three-element totally ordered set, and the other is a three-element semilattice that is not a lattice)
Semigroup_with_three_elements
Existence of certain infima or suprema of a given poset
suprema are known to exist is therefore called a join-semilattice. The dual notion is meet-semilattice. The strongest form of completeness is the existence
Completeness_(order_theory)
Example of a Semigroup
"and"), or equivalently the set {0,1} under multiplication: the only semilattice with two elements and the only non-null semigroup with zero of order
Semigroup_with_two_elements
Generalization of vector spaces from fields to rings
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Module_(mathematics)
Algebraic ring without a multiplicative identity
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Rng_(algebra)
Binary relation which never occurs in both directions
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗
Asymmetric_relation
Algebraic structure
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Finite_field
Sets with binary operations analogous to the Reidemeister moves used on knot diagrams
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Racks_and_quandles
Arrangement of hyperplanes
flag consisting of modular elements. Equivalently, the intersection semilattice of the arrangement is a supersolvable lattice, in the sense of Richard
Supersolvable_arrangement
Binary relation that relates every element to itself
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗
Reflexive_relation
Algebra with unique prime factorization
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Dedekind_domain
Order-preserving mathematical function
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Monotonic_function
Algebraic structure also called skew field
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Division_ring
Set with associative invertible operation
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Group_(mathematics)
Type of binary relation
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗
Transitive_relation
Mathematical proposition equivalent to the axiom of choice
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Zorn's_lemma
Class of mathematical orderings
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗
Well-order
Mathematical ring with well-behaved ideals
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Noetherian_ring
Algebraic structure in linear algebra
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Vector_space
Algebraic structure with addition, multiplication, and division
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Field_(mathematics)
Set of the values of a function
function is a lattice homomorphism, while the image function is only a semilattice homomorphism (that is, it does not always preserve intersections). Bijection
Image_(mathematics)
Semigroup in abstract algebra
complement of V is the meet of E and F. Since projections form a meet-semilattice, the partial isometries on Mn(C) form an inverse semigroup with the product
Semigroup_with_involution
Type of binary relation
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗
Well-founded_relation
Mathematical structure with greatest common divisors
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
GCD_domain
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Locally_finite_poset
Kind of non-classical logic
0\cdot x=x} Under these conditions, the operational frame is a join-semilattice. An operational model M {\displaystyle M} is a frame F {\displaystyle
Relevance_logic
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
chain, as either a Boolean algebra, a Heyting algebra, a lattice, or a semilattice, is subdirectly irreducible. In fact, the two-element chain is the only
Subdirectly irreducible algebra
Subdirectly_irreducible_algebra
Concept in mathematics
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Map_of_lattices
On chains and antichains in partial orders
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Dilworth's_theorem
Commutative group (mathematics)
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Abelian_group
Algebraic structure
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Principal_ideal_domain
Branch of mathematics
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Order_theory
Algebraic object with an ordered structure
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Ordered_field
Branch of algebra
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Ring_theory
Bound lattice in which every element has a complement
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Complemented_lattice
Mathematical relation inside orderings
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Covering_relation
Families of certain algebraic structures
Fennemore Normal band A band such that abca = acba Infinite Finite Fennemore Semilattice A commutative band, that is: a2 = a ab = ba Infinite Finite C&P p. 24
Special_classes_of_semigroups
Subset of a preorder that contains all larger elements
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Upper_and_lower_sets
Algebraic structure with a binary operation
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Magma_(algebra)
Partially ordered vector space, ordered as a lattice
E} a preordered vector space. Item 3 says that the preorder is a join semilattice. Because the preorder is compatible with the vector space structure,
Riesz_space
Tree node with two other nodes as descendants
in distributed computing (Bender et al. 2005). Level ancestor problem Semilattice Christianson, B.; Santanu, Dash (2015). "Try our source code for free"
Lowest_common_ancestor
Algebraic structure with an associative operation and an identity element
is the empty set). Generalizing the previous example, every bounded semilattice is an idempotent commutative monoid. In particular, any bounded lattice
Monoid
Generalization of the concept of subsequence to the case of nets
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Subnet_(mathematics)
Reflexive and transitive binary relation
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗
Preorder
Mathematical property of subsets in order theory
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Cofinal_(mathematics)
Type of algebraic structure
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Graded_ring
A congruence θ of a join-semilattice S is monomial, if the θ-equivalence class of any element of S has a largest element. We say that θ is distributive
Distributive_homomorphism
Term in the mathematical area of order theory
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Duality_(order_theory)
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Symmetric_closure
Commutative ring with no zero divisors other than zero
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Integral_domain
Generalized topological space
relation. Chu(Set, 2) realizes a wide range of "logical" structures such as semilattices, distributive lattices, complete and completely distributive lattices
Chu_space
Ring that is also a vector space or a module
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Associative_algebra
Hungarian mathematician
mathematician. Huhn's theorem on the representation of distributive semilattices is named after him. O'Connor, John J.; Robertson, Edmund F., "András
András_P._Huhn
Smallest transitive relation containing a given binary relation
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗
Transitive_closure
Ring without nonzero zero divisors
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Domain_(ring_theory)
Additionally, Scott domains appear with other names like "algebraic semilattice" in some publications. Originally, Dana Scott demanded a complete lattice
Scott_domain
Type of integral domain
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Unique_factorization_domain
Set theory concept
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Club_set
Measure of unsolvability
degrees of X and Y. Thus D {\displaystyle {\mathcal {D}}} is a join-semilattice. The least upper bound of degrees a and b is denoted a ∪ b. It is known
Turing_degree
Commutative ring with a Euclidean division
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Euclidean_domain
Mathematical concept for comparing objects
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗
Well-quasi-ordering
Binary relation over a set and itself
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Homogeneous_relation
Type of topological space in mathematics
Zbl 0522.54003 Bice, Tristan; Kubiś, Wiesław (2020). "Wallman Duality for Semilattice Subbases". arXiv:2002.05943 [math.GN]. Steen & Seebach, p. 20 Kelley
Locally_compact_space
Type of algebras, possibly non associative
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Composition_algebra
Algebraic ring that need not have additive negative elements
domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map
Semiring
Overview of and topical guide to algebraic structures
represented using three binary operations. Loop: a quasigroup with identity. Semilattice: a semigroup whose operation is idempotent and commutative. The binary
Outline of algebraic structures
Outline_of_algebraic_structures
Subset of incomparable elements
Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering
Antichain
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