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Algebraic Structures
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
An algebraic structure consists of one or more sets together with one or more operations which enable members of the sets to be combined in some way. What is important about a particular algebraic structure is that many of its properties are predictable from the characteristics of the operation or operations involved. This means that we can classify algebraic structures into families whose members have many features in common. Identification of a given algebraic structure as belonging to a particular family of structures allows us to conclude that it has the properties characteristic of all members of the family. To illustrate the point: you may know nothing about a lory. However, if you are told that it is a type of parrot, then you may reasonably assume that it has amongst its attributes all those which are characteristic of parrots. So it is with algebraic structures. If a particular structure can be identified as a ‘group' then it can be assumed to have all the properties characteristic of groups.
Algebra for Enterprise Ontology: towards analysis and synthesis of enterprise models
Published in Enterprise Information Systems, 2018
Step 1 requires the definitions for a set, operations, and axioms because an algebraic structure, also known as an algebra, is fundamentally defined by a set S (called the underlying set) and one or more operations on S that are required to obey certain axioms. The set S is literally a set of things in the scope of interest, thus, DEMO OCD. Each operation is described in the form of an n-ary operation on S (i.e., taking n elements of S as inputs and returning a single element of S). Axioms are interpreted as rules in plain English. They are usually formulated in the form of equational laws.