Complete latticeIn 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 and nonempty 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.
Lattice (order)Set whose pairs have minima and maximaY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
Heyting algebraIn mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b called implication such that (c ∧ a) ≤ b is equivalent to c ≤ (a → b).
Boolean algebra (structure)In mathematics, a Boolean algebra or Boolean lattice is a complementeddistributive 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).
Commutative propertyIn mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings.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". Background and motivationOrders are everywhere in mathematics and related fields like computer science.Associative propertyIn mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a validrule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed.
Infimum and supremumGreatest lower bound and least upper boundA set P{\displaystyle P} of real numbers (hollow and filled circles), a subset S{\displaystyle S} of P{\displaystyle P} (filled circles), and the infimum of S.{\displaystyle S.} Note that for totally ordered finite sets, the infimum and the minimum are equal.A set A{\displaystyle A} of real numbers (blue circles), a set of upper bounds of A{\displaystyle A} (red diamond and circles), and the smallest such upper bound, that is, the supremum of A{\displaystyle A} (red diamond).In mathematics, the infimum (abbreviated inf; pl.: infima) of a subsetS{\displaystyle S} of a partially ordered setP{\displaystyle P} is the greatest element in P{\displaystyle P} that is less than or equal to each element of S,{\displaystyle S,} if such an element exists.Greatest element and least elementIn mathematics, especially in order theory, the greatest element of a subset S{\displaystyle S} of a partially ordered set (poset) is an element of S{\displaystyle S} that is greater than every other element of S{\displaystyle S}.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.*As an Amazon Associate I earn from qualifying purchases.