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A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
Often it is a list of operations (actions) ordered by time, performed by a set of transactions that are executed together in the system. If the order in time between certain operations is not determined by the system, then a partial order is used.
A partially ordered set is a directed-complete partial order (dcpo) if each of its directed subsets has a supremum. (A subset of a partial order is directed if it is non-empty and every pair of elements has an upper bound in the subset.)
Ordered storage typically stores the records in order and may have to rearrange or increase the file size when a new record is inserted, resulting in lower insertion efficiency. However, ordered storage provides more efficient retrieval as the records are pre-sorted, resulting in a complexity of O ( log n ) {\displaystyle O\left(\log n ...
A database transaction symbolizes a unit of work, performed within a database management system (or similar system) against a database, that is treated in a coherent and reliable way independent of other transactions. A transaction generally represents any change in a database. Transactions in a database environment have two main purposes:
Formally, a "database" refers to a set of related data accessed through the use of a "database management system" (DBMS), which is an integrated set of computer software that allows users to interact with one or more databases and provides access to all of the data contained in the database (although restrictions may exist that limit access to particular data).
Consider a partially ordered set (X, ≤). As a first simple example, let 1 = {*} be a specified one-element set with the only possible partial ordering. There is an obvious mapping j: X → 1 with j(x) = * for all x in X. X has a least element if and only if the function j has a lower adjoint j *: 1 → X.
Many important examples of directed sets can be defined using these partial orders. For example, by definition, a prefilter or filter base is a non-empty family of sets that is a directed set with respect to the partial order and that also does not contain the empty set (this condition prevents triviality because otherwise, the empty set would ...