irreducibility
irreducibility (adj)
irreducibility
English Definitions:
irreducibility (Noun)
The quality or degree of being irreducible.
irreducibility (Noun)
Something that is irreducible.
Irreducibility
In mathematics, the concept of irreducibility is used in several ways. ⁕In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial. ⁕In representation theory, an irreducible representation is a nontrivial representation with no nontrivial proper subrepresentations. Similarly, an irreducible module is another name for a simple module. ⁕Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as irreducible over an algebraic closure. ⁕In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space. ⁕A directed graph is irreducible if, given any two vertices, there exists a directed path from the first vertex to the second. This is equivalent to the digraph being strongly connected. The term derives from the fact that a digraph is irreducible if and only if its adjacency matrix is irreducible.¹¹
irreducibility
In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence plays a central role in theories of integrative levels and of complex systems. For instance, the phenomenon of life as studied in biology is an emergent property of chemistry. In philosophy, theories that emphasize emergent properties have been called emergentism.
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"irreducibility." Kamus.net. STANDS4 LLC, 2024. Web. 28 Mar. 2024. <https://www.kamus.net/english/irreducibility>.
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