The tee (⊤), also called down tack (as opposed to the up tack) or verum is a symbol used to represent: The top element in lattice theory. A logical constant denoting a tautology in logic. The top type in type theory. In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element that is smaller than every other element of S. Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if s ≤ g, for all elements s of S. Hence, the greatest element of S is an upper bound of S that is contained within this subset. It is necessarily unique. By using ≥ instead of ≤ in the above definition, one defines the least element of S. Like upper bounds, greatest elements may fail to exist. Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative real numbers. This example also demonstrates that the existence of a least upper bound (the number 0 in this case) does not imply the existence of a greatest element either. Similar conclusions hold for least elements. A finite chain always has a greatest and a least element. A greatest element of a partially ordered subset must not be confused with maximal elements of the set, which are elements that are not smaller than any other elements. A set can have several maximal elements without having a greatest element. However, if it has a greatest element, it can't have any other maximal element. In a totally ordered set both terms coincide; it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum.[1] The dual terms are minimum and absolute minimum. Together they are called the absolute extrema. The least and greatest element of the whole partially ordered set plays a special role and is also called bottom and top, or zero (0) and unit (1), or ⊥ and ⊤, respectively. If both exists, the poset is called a bounded poset. The notation of 0 and 1 is used preferably when the poset is even a complemented lattice, and when no confusion is likely, i.e. when one is not talking about partial orders of numbers that already contain elements 0 and 1 different from bottom and top. The existence of least and greatest elements is a special completeness property of a partial order. Further introductory information is found in the article on order theory. Examples[edit] The subset ℤ has no upper bound in the poset ℝ. Let the relation "≤" on {a, b, c, d} be given by a ≤ c, a ≤ d, b ≤ c, b ≤ d. The set {a, b} has upper bounds c and d, but no least upper bound, and no greatest element. In ℚ, the set of numbers with their square less than 2 has upper bounds but no least upper bound. In ℝ, the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element. In ℝ, the set of numbers less than or equal to 1 has a greatest element, viz. 1, which is also its least upper bound. In ℝ² with the product order, the set of (x, y) with 0 < x < 1 has no upper bound. In ℝ² with the lexicographical order, this set has upper bounds, e.g. (1, 0). It has no least upper bound. In mathematics, a lattice is a partially ordered set in which every two elements have a supremum (also called a least upper bound or join) and an infimum (also called a greatest lower bound or meet). An example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions.In logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; (it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense). A formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent. Such a formula can be made either true or false based on the values assigned to its propositional variables. The double turnstile notation \vDash S is used to indicate that S is a tautology. Tautology is sometimes symbolized by "Vpq", and contradiction by "Opq". The tee symbol \top is sometimes used to denote an arbitrary tautology, with the dual symbol \bot (falsum) representing an arbitrary contradiction. Tautologies are a key concept in propositional logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. A key property of tautologies in propositional logic is that an effective method exists for testing whether a given formula is always satisfied (or, equivalently, whether its negation is unsatisfiable). The definition of tautology can be extended to sentences in predicate logic, which may contain quantifiers, unlike sentences of propositional logic. In propositional logic, there is no distinction between a tautology and a logically valid formula. In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable). The set of such formulas is a proper subset of the set of logically valid sentences of predicate logic (which are the sentences that are true in every model).Logic is the use and study of valid reasoning.[2][3] The study of logic features most prominently in the subjects of philosophy, mathematics, and computer science. Logic was studied in several ancient civilizations, including India,[4] China,[5] Persia and Greece. In the West, logic was established as a formal discipline by Aristotle, who gave it a fundamental place in philosophy. The study of logic was part of the classical trivium, which also included grammar and rhetoric. Logic was further extended by Al-Farabi who categorized it into two separate groups (idea and proof). Later, Avicenna revived the study of logic and developed relationship between temporalis and the implication. In the East, logic was developed by Buddhists and Jains. Logic is often divided into three parts: inductive reasoning, abductive reasoning, and deductive reasoning. The top type in the type theory of mathematics, logic, and computer science, commonly abbreviated as top or by the down tack symbol (⊤), is the universal type—that type which contains every possible object in the type system of interest. The top type is sometimes called the universal supertype as all other types in any given type system are subtypes of top. It is in contrast with the bottom type, or the universal subtype, which is the type containing no members at all. In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics. In type theory, every "term" has a "type" and operations are restricted to terms of a certain type. Type theory is closely related to (and in some cases overlaps with) type systems, which are a programming language feature used to reduce bugs. The types of type theory were created to avoid paradoxes in a variety of formal logics and rewrite systems and sometimes "type theory" is used to refer to this broader application. Two well-known type theories that can serve as mathematical foundations are Alonzo Church's typed λ-calculi and Per Martin-Löf's intuitionistic type theory.