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Universal Algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures. For instance, rather than considering groups or rings as the object of studythis is the subject of group theory and ring theory in universal algebra, the object of study is the possible types of algebraic structures and their relationships. Basic idea In universal algebra, an (or algebraic structure) is a set ''A'' together with a collection of operations on ''A''. Arity An ''n''- ary operation on ''A'' is a function that takes ''n'' elements of ''A'' and returns a single element of ''A''. Thus, a 0-ary operation (or ''nullary operation'') can be represented simply as an element of ''A'', or a '' constant'', often denoted by a letter like ''a''. A 1-ary operation (or '' unary operation'') is simply a function from ''A'' to ''A'', often denoted by a symbol placed in front of its argument, like ~'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum ( join) and an infimum ( meet). A conditionally complete lattice satisfies at least one of these properties for bounded subsets. For comparison, in a general lattice, only ''pairs'' of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete. Complete lattices appear in many applications in mathematics and computer science. Both order theory and universal algebra study them as a special class of lattices. Complete lattices must not be confused with complete partial orders (CPOs), a more general class of partially ordered sets. More specific complete lattices are complete Boolean algebras and complete Heyting algebras (locales). Formal definition A ''complete lattice'' is a partially ordered set (''L'', ≤) such that every subset ''A'' of ''L'' has both a greatest lower bound (the infimum, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be definable set, defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Theory
Order 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. Background and motivation 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.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems in general (although one usually is also interested in the actual difference of two numbers, which is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater than (denoted by and , respectively the less-than sign, less-than and greater-than sign, greater-than signs). Notation There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equality is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' or ''a'' ≦ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitary Relation
In mathematics, a finitary relation over a sequence of sets is a subset of the Cartesian product ; that is, it is a set of ''n''-tuples , each being a sequence of elements ''x''''i'' in the corresponding ''X''''i''. Typically, the relation describes a possible connection between the elements of an ''n''-tuple. For example, the relation "''x'' is divisible by ''y'' and ''z''" consists of the set of 3-tuples such that when substituted to ''x'', ''y'' and ''z'', respectively, make the sentence true. The non-negative integer ''n'' that gives the number of "places" in the relation is called the ''arity'', ''adicity'' or ''degree'' of the relation. A relation with ''n'' "places" is variously called an ''n''-ary relation, an ''n''-adic relation or a relation of degree ''n''. Relations with a finite number of places are called ''finitary relations'' (or simply ''relations'' if the context is clear). It is also possible to generalize the concept to ''infinitary relations'' with Sequence, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Conjunction
In logic, mathematics and linguistics, ''and'' (\wedge) is the Truth function, truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as \wedge or \& or K (prefix) or \times or \cdot in which \wedge is the most modern and widely used. The ''and'' of a set of operands is true if and only if ''all'' of its operands are true, i.e., A \land B is true if and only if A is true and B is true. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English language, English "Conjunction (grammar), and"; * In programming languages, the Short-circuit evaluation, short-circuit and Control flow, control structure; * In set theory, Intersection (set theory), intersection. * In Lattice (order), lattice theory, logical conjunction (Infimum and supremum, greatest lower bound). Notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary connective \lor can be used to join the two atomic formulas P and Q, rendering the complex formula P \lor Q . Common connectives include negation, disjunction, conjunction, implication, and equivalence. In standard systems of classical logic, these connectives are interpreted as truth functions, though they receive a variety of alternative interpretations in nonclassical logics. Their classical interpretations are similar to the meanings of natural language expressions such as English "not", "or", "and", and "if", but not identical. Discrepancies between natural language connectives and those of classical logic have motivated nonclassical approaches to natural language meaning as well as approaches which pair a classi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existential Quantification
Existentialism is a family of philosophy, philosophical views and inquiry that explore the human individual's struggle to lead an Authenticity (philosophy), authentic life despite the apparent Absurdity#The Absurd, absurdity or incomprehensibility of existence. In examining meaning of life, meaning, purpose, and value (ethics), value, existentialist thought often includes concepts such as existential crisis, existential crises, Angst#Existentialist angst, angst, courage, and freedom. Existentialism is associated with several 19th- and 20th-century European philosophers who shared an emphasis on the human subject, despite often profound differences in thought. Among the 19th-century figures now associated with existentialism are philosophers Søren Kierkegaard and Friedrich Nietzsche, as well as novelist Fyodor Dostoevsky, all of whom critiqued rationalism and concerned themselves with the problem of meaning (philosophy), meaning. The word ''existentialism'', however, was not coin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Quantification
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable. It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("", "", or sometimes by "" alone). Universal quantification is distinct from ''existential'' quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain. Quantification in general is covered in the article on quantification (logic). The universal quantifier is en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantification (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first-order formula \forall x P(x) expresses that everything in the domain satisfies the property denoted by P. On the other hand, the existential quantifier \exists in the formula \exists x P(x) expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable. The most commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic: each can be defined in terms of the other using negation. They can also be used to define more complex quantifiers, as in the formula \neg \exists x P(x) which expresses that nothing ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variety (universal Algebra)
In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called ''finitary algebraic categories''. A ''covariety'' is the class of all coalgebraic structures of a given signature. Terminology A variety of algebras should not be confused with an algebraic variety, which means a set of solutions to a system of polynomial equations. They are formally quite distinct and their theories have little in common. The term "variety of algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
