Boolean algebraIn mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and the negation (not) denoted as ¬.
Lattice (order)A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection.
Order theoryOrder theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.
Distributive latticeIn mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets. As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra.
Limit-preserving function (order theory)In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i.e. certain suprema or infima. Roughly speaking, these functions map the supremum/infimum of a set to the supremum/infimum of the image of the set. Depending on the type of sets for which a function satisfies this property, it may preserve finite, directed, non-empty, or just arbitrary suprema or infima. Each of these requirements appears naturally and frequently in many areas of order theory and there are various important relationships among these concepts and other notions such as monotonicity.
Complete Heyting algebraIn mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the of three different ; the category CHey, the category Loc of locales, and its , the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras.
Galois connectionIn mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois. A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets.
ToposIn mathematics, a topos (USˈtɒpɒs, UKˈtoʊpoʊs,_ˈtoʊpɒs; plural topoi ˈtɒpɔɪ or ˈtoʊpɔɪ, or toposes) is a that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic. The mathematical field that studies topoi is called topos theory.
Boolean algebra (structure)In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).
Stone dualityIn mathematics, there is an ample supply of categorical dualities between certain of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics.
Ideal (order theory)In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory. A subset I of a partially ordered set is an ideal, if the following conditions hold: I is non-empty, for every x in I and y in P, y ≤ x implies that y is in I (I is a lower set), for every x, y in I, there is some element z in I, such that x ≤ z and y ≤ z (I is a directed set).
Completeness (order theory)In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions of completeness exist. The motivation for considering completeness properties derives from the great importance of suprema (least upper bounds, joins, "") and infima (greatest lower bounds, meets, "") to the theory of partial orders.
Distributivity (order theory)In the mathematical area of order theory, there are various notions of the common 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 concept can in fact reasonably be generalized to semilattices as well. Probably the most common type of distributivity is the one defined for lattices, where the formation of binary suprema and infima provide the total operations of join () and meet ().
Cartesian closed categoryIn , a is Cartesian closed if, roughly speaking, any morphism defined on a of two can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by , whose internal language, linear type systems, are suitable for both quantum and classical computation.
Pointless topologyIn mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this approach it becomes possible to construct topologically interesting spaces from purely algebraic data. The first approaches to topology were geometrical, where one started from Euclidean space and patched things together.
Truth tableA truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid.
Spectral spaceIn mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos. Let X be a topological space and let K(X) be the set of all compact open subsets of X. Then X is said to be spectral if it satisfies all of the following conditions: X is compact and T0. K(X) is a basis of open subsets of X. K(X) is closed under finite intersections. X is sober, i.e.
Double negationIn propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition A is logically equivalent to not (not-A), or by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation. Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic, but it is disallowed by intuitionistic logic.
Intuitionistic logicIntuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L.
Tautology (logic)In mathematical logic, a tautology (from ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of the ball. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement.
