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Join and meet

In lattice theory, a branch of within , the join and meet are fundamental binary operations defined on a (poset) where every pair of elements has a least upper bound and a greatest lower bound, respectively. The join of elements a and b, denoted a \vee b, is the smallest element greater than or equal to both a and b, serving as the least upper bound; conversely, the meet, denoted a \wedge b, is the largest element less than or equal to both, acting as the greatest lower bound. These operations characterize a lattice, a poset equipped with joins and meets for all finite subsets, enabling the study of algebraic structures that unify concepts from , , and . Originating in the late with Richard Dedekind's work on ideal theory and partially ordered sets, lattice theory was formalized in and 1940s by and others, who recognized joins and meets as operations satisfying properties like commutativity (a \vee b = b \vee a), associativity ((a \vee b) \vee c = a \vee (b \vee c)), and (a \vee (a \wedge b) = a). In complete lattices, joins and meets extend to arbitrary subsets, forming suprema (\bigvee) and infima (\bigwedge), which underpin applications in topology, computer science (e.g., dataflow analysis), and formal concept analysis. Common examples illustrate their utility: in the power set lattice of subsets ordered by inclusion, the join is set union and the meet is intersection; under divisibility on positive integers, the join is the least common multiple (LCM) and the meet is the greatest common divisor (GCD); and in Boolean algebras, they correspond to logical OR and AND, respectively. More specialized lattices, such as those of subgroups or vector subspaces, exhibit additional properties like modularity, where if a \leq c, then a \vee (b \wedge c) = (a \vee b) \wedge c, distinguishing them from general lattices. These concepts extend to infinite structures and varieties, influencing fields such as universal algebra.

Definitions

Partial order approach

In order theory, a partially ordered set (poset) consists of a set P together with a binary relation \leq on P that is reflexive (for all x \in P, x \leq x), antisymmetric (for all x, y \in P, if x \leq y and y \leq x, then x = y), and transitive (for all x, y, z \in P, if x \leq y and y \leq z, then x \leq z). This relation provides a foundational structure for comparing elements, where some pairs may be comparable (one precedes the other) while others are incomparable. Within a poset, the meet of two elements a, b \in P, denoted a \wedge b, is defined as their greatest lower bound, provided it exists. This means a \wedge b is an element z \in P such that:
  • z \leq a and z \leq b, and
  • for all w \in P, if w \leq a and w \leq b, then w \leq z.
The meet thus satisfies the universal property of being the largest element below both a and b. Dually, the join of a and b, denoted a \vee b, is their least upper bound, provided it exists. This is an element z \in P such that:
  • a \leq z and b \leq z, and
  • for all w \in P, if a \leq w and b \leq w, then z \leq w.
The join is therefore the smallest element above both a and b. Not every pair of elements in a poset necessarily possesses a meet or a join, as the required bounds may fail to exist./01:Generative_Effects-_Orders_and_Adjunctions/1.02:_Meets_and_Joins) However, whenever a meet or join does exist for a given pair, it is unique: if z and z' are both greatest lower bounds of a and b, then z \leq z' and z' \leq z, so z = z' by antisymmetry of the order. The same uniqueness holds for joins by a argument./01:Generative_Effects-_Orders_and_Adjunctions/1.02:_Meets_and_Joins)

Universal algebra approach

In , a is defined as a set L equipped with two binary operations, meet (denoted \wedge) and join (denoted \vee), that satisfy the following axioms for all a, b, c \in L:
  • Commutativity: a \wedge b = b \wedge a and a \vee b = b \vee a.
  • Associativity: (a \wedge b) \wedge c = a \wedge (b \wedge c) and (a \vee b) \vee c = a \vee (b \vee c).
  • Idempotence: a \wedge a = a and a \vee a = a.
  • Absorption: a \wedge (a \vee b) = a and a \vee (a \wedge b) = a.
These identities characterize the purely in terms of algebraic operations, without reference to an underlying . The pair of operations (\wedge, \vee) thus forms a (L, \wedge, \vee), which belongs to the of s in the sense of , where varieties are classes of algebras defined by equational laws. This algebraic perspective allows s to be studied alongside other structures like groups or rings, emphasizing identities that hold universally within the class. From these operations, a partial order can be induced on L by defining a \leq b a \wedge b = a (equivalently, a \vee b = b); this is reflexive, antisymmetric, and transitive, recovering the order-theoretic view of lattices. This algebraic formulation of lattices originated in the work of Garrett Birkhoff during the 1930s and 1940s, particularly in his seminal 1940 monograph Lattice Theory, which systematized the subject and integrated it into broader universal algebraic frameworks.

Equivalence of approaches

The equivalence between the partial order and universal algebra approaches to defining lattices establishes that these perspectives describe the same structures, thereby unifying lattice theory across order-theoretic and algebraic frameworks. Specifically, given a set L equipped with binary operations \wedge and \vee satisfying the lattice axioms—commutativity (a \wedge b = b \wedge a, a \vee b = b \vee a), associativity (a \wedge (b \wedge c) = (a \wedge b) \wedge c, a \vee (b \vee c) = (a \vee b) \vee c), absorption (a \wedge (a \vee b) = a, a \vee (a \wedge b) = a), and idempotence (a \wedge a = a, a \vee a = a)—one can induce a partial order \leq on L by defining a \leq b if and only if a \wedge b = a (equivalently, a \vee b = b). Under this order, (L, \leq) is a partially ordered set (poset) in which every pair of elements has a greatest lower bound (glb, or meet) given by \wedge and a least upper bound (lub, or join) given by \vee. To see that \leq is a partial , reflexivity follows from : a \wedge a = a, so a \leq a. Antisymmetry holds because if a \leq b and b \leq a, then a = a \wedge b = b \wedge a = b by and commutativity. For , assume a \leq b and b \leq c; then a \wedge b = a and b \wedge c = b. Associativity yields a \wedge c = (a \wedge b) \wedge c = a \wedge (b \wedge c) = a \wedge b = a, so a \leq c. Next, \wedge acts as the glb of a and b: it is a lower bound since (a \wedge b) \wedge a = a \wedge (a \wedge b) = a \wedge b (so a \wedge b \leq a) and similarly a \wedge b \leq b, and it is greatest because any x \leq a and x \leq b satisfies x \wedge (a \wedge b) = (x \wedge a) \wedge b = x \wedge b = x (so x \leq a \wedge b). Dually, \vee is the lub of a and b: it is an upper bound (a \leq a \vee b, b \leq a \vee b) and least because any upper bound y satisfies (a \vee b) \vee y = y \vee (a \vee b) = (y \vee a) \vee b = y \vee b = y (so a \vee b \leq y). Thus, the yields an order-theoretic . Conversely, suppose (L, \leq) is a poset in which every pair of elements has a glb (meet) and lub (join). Define a \wedge b as the glb of a and b, and a \vee b as the lub. These operations satisfy the lattice axioms: and commutativity are immediate from the definitions of glb and lub; associativity follows because the glb (lub) of three elements is independent of grouping, as \mathrm{glb}(a, \mathrm{glb}(b, c)) = \mathrm{glb}(\mathrm{glb}(a, b), c) by the universal property of glb; holds since \mathrm{glb}(a, \mathrm{lub}(a, b)) = a (as a \leq \mathrm{lub}(a, b) and a is a lower bound for itself and the lub). Moreover, these operations are unique: in the order-theoretic setting, the meet and join of any pair are uniquely determined as the glb and lub, while in the algebraic setting, the induced partial order is the unique order compatible with the operations as infima and suprema. This bidirectional correspondence ensures that every algebraic is order-isomorphic to an order-theoretic one, and vice versa, with the operations coinciding under the induced order. The equivalence arises from the laws, which bridge the algebraic identities to order relations, allowing seamless translation between the frameworks.

Core Properties

Binary operation laws

In lattice theory, the join (∨) and meet (∧) operations on a form that satisfy the laws of commutativity, associativity, and , which are essential for defining the of a . These properties ensure that joins and meets behave consistently as supremum and infimum operations, respectively, allowing lattices to model various ordered structures in . Commutativity holds for both operations, meaning the result is independent of the order of the arguments:
a \wedge b = b \wedge a
a \vee b = b \vee a.
This follows from the inherent in the definitions of meet as the greatest lower bound and join as the least upper bound of the pair \{a, b\}, since swapping a and b yields the same bounds. Associativity ensures that the grouping of multiple operands does not affect the outcome:
(a \wedge b) \wedge c = a \wedge (b \wedge c)
(a \vee b) \vee c = a \vee (b \vee c).
In the order-theoretic view, this arises because the infimum (meet) of three elements is the greatest element bounded above by all three, regardless of pairwise grouping, and similarly for the supremum (join). Idempotence means that applying the to an with itself yields the unchanged:
a \wedge a = a
a \vee a = a.
This property derives from the fact that a is both the greatest lower bound and least upper bound of the singleton set \{a\}. Together, these laws enable the unambiguous extension of join and meet to finite n-ary operations on any finite subset of the . For instance, the meet a \wedge b \wedge c can be computed as (a \wedge b) \wedge c or a \wedge (b \wedge c), yielding the same infimum of \{a, b, c\}, with similar unambiguity for joins.

Absorption and distributivity

In lattice theory, the absorption laws relate the join and meet operations, stating that for all elements a and b in a L, a \wedge (a \vee b) = a and its dual a \vee (a \wedge b) = a. These identities hold in every , as they follow directly from the order-theoretic definitions of join and meet as the least upper bound and greatest lower bound, respectively. Specifically, since a \leq a \vee b, the meet a \wedge (a \vee b) is the greatest lower bound of a and an element greater than or equal to a, yielding a. The dual argument applies by , interchanging joins and meets (or ). Distributivity provides a further interaction between join and meet, requiring that for all a, b, c \in L, a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c) and its dual a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c). A lattice satisfying both identities is called distributive; these laws ensure that the operations mimic the distribution in numerical algebra, enabling representations like Birkhoff's theorem for finite distributive lattices as rings of sets. Distributivity implies modularity (defined below) but is strictly stronger, as verified by checking the modular law as a special case when a \leq b. Modularity weakens distributivity while still capturing significant structure, defined by the modular law: for all a, b, c \in L with a \leq b, a \vee (b \wedge c) = b \wedge (a \vee c). Equivalently, in full generality, (a \vee c) \wedge (a \vee b) = a \vee ((a \vee b) \wedge c), which holds if and only if the lattice embeds no sublattice isomorphic to the 5-element non-modular lattice N_5 (the pentagon). Every distributive lattice is modular, but not conversely; for instance, the lattice of subspaces of a vector space over a division ring is modular but typically not distributive unless the dimension is at most 1. Lattices failing distributivity exhibit sublattices isomorphic to either the 3-element modular non-distributive lattice M_3 (the diamond, with three atoms over a bottom element) or the non-modular N_5. A finite lattice is distributive if and only if it contains neither as a sublattice, providing a forbidden-substructure characterization that distinguishes lattice varieties.

Generalizations

Finite subsets

In lattice theory, the binary join and meet operations extend naturally to finite non-empty subsets through . For a finite S = \{a_1, a_2, \dots, a_n\} \subseteq L of a L, the meet of S is defined as \bigwedge S = a_1 \wedge a_2 \wedge \dots \wedge a_n, obtained by successively applying the binary meet operation. This iterated operation yields a unique result, independent of the grouping or order of the elements, due to the associativity and commutativity of the meet in lattices. Similarly, the join of S is \bigvee S = a_1 \vee a_2 \vee \dots \vee a_n, iterated via the binary join, which is also associative and commutative, ensuring uniqueness. The meet \bigwedge S serves as the greatest lower bound of S in the lattice order: it is a lower bound for every element in S, and any other lower bound is less than or equal to \bigwedge S. Thus, for any u \in L such that u \leq a_i for all i = 1, \dots, n, it follows that u \leq \bigwedge S. Analogously, \bigvee S is the least upper bound of S, satisfying a_i \leq \bigvee S for all i and \bigvee S \leq v for any upper bound v of S. Lattices guarantee the existence of both joins and meets for every finite non-empty , as the operations suffice to construct them iteratively without requiring additional structure. This finitariness aligns with the algebraic characterization of lattices as posets equipped with associative, commutative, and idempotent operations satisfying laws. A key property is monotonicity: if S \subseteq T are finite non-empty subsets of L, then \bigwedge T \leq \bigwedge S, since adding elements to the subset can only decrease (or maintain) the greatest lower bound. The same holds dually for joins: \bigvee S \leq \bigvee T. These properties preserve the order structure while extending the binary operations to finite collections.

Infinite joins and meets

In a complete lattice, every subset S possesses both a meet \bigwedge S, defined as the greatest lower bound (infimum) of S, and a join \bigvee S, defined as the least upper bound (supremum) of S. This extends the binary operations to arbitrary collections, ensuring the existence of these bounds even for infinite subsets, unlike in general lattices where only finite subsets are guaranteed such bounds. Finite meets and joins arise as special cases when S is finite. In general partially ordered sets (posets), arbitrary meets and joins may fail to exist; for instance, the poset of rational numbers \mathbb{Q} under the usual order \leq forms a lattice for finite subsets but lacks a supremum for the bounded subset \{x \in \mathbb{Q} \mid x^2 < 2\}, as any potential least upper bound in \mathbb{Q} would contradict the irrationality of \sqrt{2}. However, if every subset of a poset has a meet, then it automatically has all joins, rendering the poset a complete lattice; specifically, for any subset A, the join \bigvee A equals the meet of all upper bounds of A. Continuous lattices, a subclass of complete lattices, exhibit continuity properties, such as upper continuity: for any D (where every finite has an upper bound in D) with supremum \bigvee D, and any element a, it holds that a \wedge \bigvee D = \bigvee \{a \wedge d \mid d \in D\}. Dually, lower continuity ensures a \vee \bigwedge D = \bigwedge \{a \vee d \mid d \in D\} for directed D. For directed unions, this aligns with \bigwedge (\bigcup_i S_i) = \bigwedge_i (\bigwedge S_i) under appropriate conditions in continuous lattices. The Dedekind-MacNeille provides a way to embed any poset into a , constructing the smallest containing the poset as a sublattice by taking the under the operator that maps subsets to the meet of their upper bounds (or dually). This preserves existing finite joins and meets while adding infinite ones as needed, facilitating the study of order structures lacking completeness. For countable subsets, a \sigma-complete lattice is a lattice in which countable joins and meets exist. Such lattices are essential in contexts like measure theory, where countable operations suffice.

Examples

Abstract structures

In abstract algebra, join and meet operations are fundamental to lattice structures, where the join (∨) serves as the least upper bound and the meet (∧) as the greatest lower bound for any pair of elements in a partially ordered set. Bounded lattices extend this by including a top element ⊤, defined as the join of the empty set, since every element bounds the empty set from above, making ⊤ the least such bound; similarly, the bottom element ⊥ is the meet of the empty set, as it is the greatest lower bound for the empty collection. Distributive lattices satisfy the property that meets distribute over joins and vice versa, providing a framework where operations align with intuitive set-theoretic behaviors. This structure exemplifies how abstract joins and meets capture supremal and infimal elements without relying on specific computations. Modular lattices generalize distributive ones by satisfying a weaker modularity condition, illustrated by the lattice of subspaces of a vector space under inclusion, where the meet ∧ is the intersection of subspaces and the join ∨ is their sum (the span of their union). Boolean algebras represent a special case of complemented distributive lattices, equipped with a complement operation ¬ for each element a such that a ∨ ¬a = ⊤, ensuring every element pairs with its "opposite" to reach the top element. Semilattices provide a minimal setting for these operations: a meet-semilattice possesses all finite meets but not necessarily joins, focusing solely on infima, while a join-semilattice emphasizes suprema.

Concrete mathematical instances

In the divisor lattice, the set of positive integers ordered by divisibility forms a distributive lattice where the meet of two elements a and b is their greatest common divisor \gcd(a, b), and the join is their least common multiple \operatorname{lcm}(a, b). For instance, considering a = 12 and b = 18, we have \gcd(12, 18) = 6 as the meet, since 6 divides both and is the largest such integer, and \operatorname{lcm}(12, 18) = 36 as the join, since 36 is the smallest integer divisible by both. This structure highlights how joins and meets capture common divisors and multiples in number theory. The closed interval [0, 1] of real numbers, equipped with the standard order \leq, constitutes a bounded lattice where the meet of two elements x and y is \min(x, y) and the join is \max(x, y). For example, with x = 0.4 and y = 0.8, the meet is \min(0.4, 0.8) = 0.4 and the join is \max(0.4, 0.8) = 0.8; the bottom element is 0 and the top is 1. This example illustrates a chain (totally ordered set) that is a lattice, with operations aligning directly with the order. Not all partially ordered sets are lattices, as some lack joins or meets for certain pairs. A simple non-lattice poset is an of two incomparable elements a and b, where no relation holds between them; here, \{a, b\} has neither a greatest lower bound (the set of common lower bounds is empty) nor a least upper bound (the set of common upper bounds is empty). This demonstrates a "gap" in the structure, preventing the poset from being a lattice. For matrix orders, consider the set of $2 \times 2 real matrices under the componentwise partial order, where A \leq B A_{ij} \leq B_{ij} for all entries i, j. This forms a product lattice (as a of copies of the real ), with the meet given by componentwise minimum and the join by componentwise maximum. For matrices A = \begin{pmatrix} 1 & 3 \\ 2 & 0 \end{pmatrix} and B = \begin{pmatrix} 4 & 1 \\ -1 & 5 \end{pmatrix}, the meet is \begin{pmatrix} \min(1,4) & \min(3,1) \\ \min(2,-1) & \min(0,5) \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ -1 & 0 \end{pmatrix} and the join is \begin{pmatrix} \max(1,4) & \max(3,1) \\ \max(2,-1) & \max(0,5) \end{pmatrix} = \begin{pmatrix} 4 & 3 \\ 2 & 5 \end{pmatrix}. This extends scalar lattices to multidimensional settings via componentwise operations. Extending to infinite cases, the extended real numbers \overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}, ordered by the usual extension where -\infty < r < +\infty for all real r, form a with meet as and join as , featuring -\infty as the bottom element \bot and +\infty as the element \top. For the unbounded increasing x_n = n, the join is \bigvee x_n = +\infty, while for the decreasing y_n = -n, the meet is \bigwedge y_n = -\infty; finite pairs follow the real /max, such as \min(2, 5) = 2 and \max(2, 5) = 5. This ensures all subsets have suprema and infima, making it order complete.

Applications

In order theory

In order theory, joins and meets play a fundamental role in constructing free lattices within varieties of lattices, particularly those generated by partially ordered sets (posets). The free lattice generated by a poset P is the smallest lattice containing P as a subposet, obtained by formally adjoining all necessary joins and meets while preserving the order relations in P. This construction ensures the universal property: any order-preserving map from P to another lattice extends uniquely to a lattice homomorphism from the free lattice to that lattice. Such free lattices are key to understanding varieties of lattices, as they provide generators and relations that define the , allowing for the study of embeddings and representations of posets within lattice-theoretic frameworks. Heyting algebras represent a significant extension of lattices via joins and meets, serving as the algebraic models for intuitionistic propositional logic. In a Heyting algebra, the meet operation \wedge corresponds to , while the join \vee corresponds to disjunction; additionally, it includes a relative pseudocomplement operation defining a \to b = \bigvee \{ x \mid a \wedge x \leq b \}, which captures the intuitionistic notion of entailment without the law of excluded middle. This structure arises as a bounded distributive lattice equipped with this pseudocomplement, enabling the interpretation of intuitionistic proofs as elements within the algebra. Heyting algebras thus extend order theory by providing a semantic foundation for non-classical logics, where joins and meets mediate between algebraic order and logical inference. The concept of complements in , particularly orthocomplements, further enriches order-theoretic extensions involving joins and meets. An orthocomplemented is a bounded where each a has an orthocomplement a' satisfying a \vee a' = \top and a \wedge a' = \bot, with the orthocomplementation being an order-reversing (i.e., (a')' = a and if a \leq b then b' \leq a'). This structure generalizes algebras within the lattice framework, allowing for the study of quantum-like logics where complements behave orthogonally under joins and meets. Orthocomplemented lattices bridge classical and non-classical order structures, facilitating applications in duality theories. Stone duality provides a profound connection between joins and meets in —a special case of complemented distributive —and topological spaces, addressing representational gaps in . Specifically, Stone's representation theorem establishes that every is isomorphic to the algebra of clopen sets in a compact, totally disconnected (a ), where joins and meets correspond to unions and intersections of these sets. This duality, which aligns the lattice operations with set-theoretic constructions, reveals how , equipped with complements via joins and meets, dualize to , enabling topological interpretations of algebraic order. In the broader context of , this duality highlights how joins and meets in complemented varieties cover gaps between algebraic and spatial representations. Birkhoff's representation theorem addresses historical gaps in lattice theory by linking distributive lattices—where joins and meets satisfy and distributivity laws—to posets of order ideals. The theorem states that every finite distributive lattice is isomorphic to the lattice of order ideals of the poset formed by its join-irreducible elements, with joins and meets induced by unions and intersections of these ideals. This representation recovers the lattice structure from its irreducible components, providing a foundational tool for decomposing complex orders into simpler poset-based building blocks. By lattices into ideal lattices, the theorem extends , clarifying the interplay between joins, meets, and ideal completions in algebraic varieties.

In computer science

In , joins and meets play a central role in through , where they facilitate the over-approximation of concrete program states to ensure decidability. In this framework, the concrete semantics of a program is modeled over a of sets of states, and an abstract domain is defined as another that approximates these sets conservatively. The join operation in the abstract lattice computes the least upper bound of abstract values, effectively merging information from multiple execution paths to over-approximate the union of possible states, while the concretization function ensures soundness by mapping back to the concrete domain. This approach, pioneered by Patrick and Radhia Cousot, allows for the iterative computation of fixed points representing program behaviors, enabling analyses like constant propagation or pointer analysis. Lattice-based extends these ideas to compiler optimizations, where meets and joins model the flow of information across program graphs. For forward analyses such as reaching definitions, the join operation combines abstract states from incoming edges to approximate the of possible values at a program point, ensuring monotonicity for fixed-point convergence. Conversely, in backward analyses like live variables, the meet operation approximates the of outgoing states, providing a lower bound on liveness information. This lattice-theoretic structure guarantees the existence of least fixed points under finite height or widening, as formalized in Gary Kildall's unified framework, which underpins modern optimizing compilers by enabling efficient propagation of attributes like types or aliases. In for , Scott domains—algebraic complete partial orders (cpos) equipped with joins for directed suprema—model the approximation of computational processes, particularly recursive definitions. Here, the least fixed point of a interpreting a recursive program construct is computed as the join of the omega-chain generated by iterating from the bottom element, capturing the limit of finite approximations in a way that aligns with operational intuitions of . This ensures that domains form cartesian closed categories suitable for higher-order functions, providing a mathematical for the semantics of languages like PCF, as detailed in the comprehensive treatment by Samson Abramsky and Achim Jung. For finite s, which arise in applications like decision procedures or solving, algorithms for joins and meets rely on representations such as Hasse diagrams or Dedekind-MacNeille completions. Basic join computation involves finding the least upper bound by traversing the covering relations or using transitive closures, with linear in the size for sparse representations; more advanced methods employ s or implicational bases for distributive cases. The scale of such computations is illustrated by the rapid growth of s, which count the functions on n variables and thus the distributive s on n generators, with the ninth computed via symmetry-reduced requiring significant resources. These techniques are implemented in systems like the LATTICES package in for . In , fuzzy lattices extend classical s to handle uncertainty by taking truth values in a like the unit interval [0,1] with as meet and max as join, enabling graded reasoning over imprecise data. L-fuzzy sets, where membership degrees form a lattice-ordered structure, model in knowledge representation, such as in fuzzy expert systems or approximate , by propagating joins for disjunctive combinations and meets for conjunctive ones. This framework, introduced by Joseph Goguen, underpins applications in for handling noisy inputs, like in fuzzy decision trees or hybrids, where lattice operations quantify degrees of belief without binary crispness.

References

  1. [1]
    https://math.libretexts.org/Bookshelves/Applied_Mathematics/Seven_Sketches_in_Compositionality%3A_An_Invitation_to_Applied_Category_Theory_(Fong_and_Spivak)/01%3A_Generative_Effects_-_Orders_and_Adjunctions/1.02%3A_Meets_and_Joins
  2. [2]
    None
    Below is a merged response that consolidates all the information from the provided summaries into a single, comprehensive summary. To retain as much detail as possible, I will use a structured format with tables where appropriate (e.g., for definitions, properties, examples, and historical context) and narrative text for the introduction summary and URLs. This ensures a dense yet clear representation of all the data.
  3. [3]
    [PDF] Notes on Lattice Theory J. B. Nation University of Hawaii
    Theorem 2.4 is as follows. Theorem 2.5. If L is a complete meet semilattice, then L is a complete lattice with the join operation defined by. _A ...
  4. [4]
    partial order in nLab
    Jun 14, 2025 · A partial order on a set is a way of ordering its elements to say that some elements precede others, but allowing for the possibility that two elements may be ...
  5. [5]
    AMS eBooks: Colloquium Publications
    Lattice Theory. About this Title. Garrett Birkhoff. Publication: Colloquium Publications Publication Year: 1940; Volume 25 ISBNs: 978-0-8218-1025-5 (print); ...
  6. [6]
    General Lattice Theory
    **Summary of Content on Equivalence of Order and Algebraic Definitions of Lattices**
  7. [7]
    [PDF] Chapter 5. Lattices, closure operators, and Galois connections.
    Recall that an element x is said to be idempotent with respect to a binary operation * if x*x = x. The binary operation * itself is often called idempotent if ...
  8. [8]
    lattices - MathStructures - Chapman University
    A \emph{lattice} is a structure L=⟨L,∨,∧⟩ L = ⟨ L , ∨ , ∧ ⟩ , where ∨ ∨ and ∧ ∧ are infix binary operations called the \emph{join} and \emph{meet}, ...
  9. [9]
    modular lattice in nLab
    ### Summary of Modular Lattice from nLab
  10. [10]
    [PDF] 9. Modular and Semimodular Lattices
    A strongly atomic, algebraic lattice that is both upper and lower semimodular is modular. (See Theorem 3.7 of [3]; you are asked to prove the finite dimensional.
  11. [11]
    [PDF] Universal algebra and lattice theory Week 3 The distributive and ...
    Sep 17, 2020 · There are exactly two nondistributive lattices of order 5, which we call M3 and N5. We have that M3 is modular and N5 is not. M3. N5. Page 9 ...
  12. [12]
    [PDF] Finite Join and Finite Meet, and Dual Lattices - Formalized ...
    Summary. The concepts of finite join and finite meet in a lattice are introduced. Some properties of the finite join are proved. After intro-.
  13. [13]
    [PDF] Lattice Theory Lecture 1 Basics - nmsu math
    Theorem If every subset of a poset L has a meet, then every subset of L has a join, hence L is a complete lattice. ... Exercise What are joins and meets in each ...
  14. [14]
    Practical Foundations of Mathematics - Paul Taylor
    3.2 Meets, Joins and Lattices · DEFINITION 3.2. · Since finitary meets and joins are characterised by the same laws and each alone can uniquely determine the ...
  15. [15]
    [PDF] Chapter 2: Numbers - UC Davis Math
    Example 2.16. Define A ⊂ Q by. A = x ∈ Q : x2 < 2 . Then A is bounded from above by every M ∈ Q+ such that M2 > 2. Nevertheless,. A has no supremum in Q because.
  16. [16]
    sigma-complete lattice in nLab
    Jan 20, 2025 · for all elements x ∈ L x \in L and sequences s : ℕ → L s:\mathbb{N} \to L , if s ( n ) ≤ x s(n) \leq x for all natural numbers n ∈ ℕ n \in ...
  17. [17]
    [PDF] Lattice theory - Stanford Concurrency Group
    A semilattice (X, ∨) is a pair consisting of a set X and a binary operation ∨ which is associative, commutative, and idempotent. This is our first example of an ...
  18. [18]
    [PDF] Math 222A
    In a vector space V , the set of subspaces, ordered by inclusion, is a lattice. What are the meet and join operations, in more familiar terms? P ro blem E- ...
  19. [19]
    [PDF] Meet and join matrices in the poset of exponential divisors
    Thus (Z+, |) = (Z+, GCD, LCM) is a lattice, known as the divisor lattice. For a general account of lattices, see [1,3,14,19]. Let n > 1 be a positive ...
  20. [20]
    [PDF] Introduction to Lattices and Order Second edition BA Davey
    Introduction to Lattices and Order. Second edition. B.A. Davey. La Trobe University. H. A. Priestley. University of Oxford. CAMBRIDGE. UNIVERSITY PRESS. Page 2 ...
  21. [21]
    [PDF] posets.pdf
    A simple example of a poset which is not a lattice is the poset r r r r. @. @ . In a lattice, we denote the l.u.b. of {x,y} by x∨y, and the g.l.b. by x∧y.
  22. [22]
    Sublattices of product spaces: Hulls, representations and counting
    The Cartesian product of lattices is a lattice, called a product space, with componentwise meet and join operations. A sublattice of a lattice L is a subset ...
  23. [23]
    Extended Real Line - an overview | ScienceDirect Topics
    The extended real line is defined as the real number system augmented by the elements −∞ and +∞, where it is ordered such that −∞ < r < +∞ for all real numbers ...
  24. [24]
    [PDF] A fundamental non-classical logic - arXiv
    Apr 4, 2023 · Not every pseudocomplemented lattice (L, ¬) is a Heyting algebra, as Heyting algebras require a relative pseudocomplementation → such that ...
  25. [25]
    Abstract interpretation: a unified lattice model for static analysis of ...
    Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. Authors: Patrick Cousot.
  26. [26]
    [PDF] Domain Theory - Achim Jung
    This text is based on the chapter Domain Theory in the Handbook of Logic in Com- puter Science, volume 3, edited by S. Abramsky, Dov M. Gabbay, and T. S. E..
  27. [27]
    [PDF] algorithms for finite, finitely presented and free lattices
    Abstract. In this talk we will present and analyze the efficiency of various algorithms in lattice theory. For finite lattices this will include.Missing: Dedekind | Show results with:Dedekind<|control11|><|separator|>
  28. [28]
    [PDF] A computation of the ninth Dedekind Number - arXiv
    Apr 3, 2023 · In this article, we present an algorithm to compute the th Dedekind Number. The key aspects are the use of matrix multiplication and symmetries ...