/fɪˈnaɪtli ˈdʒɛnəreɪtɪd ˈstrʌkʧər/
"Finitely generated structure" refers to a mathematical structure, particularly in algebra or topology, that can be generated by a finite set of elements. This term is often used in discussions related to groups, rings, or vector spaces, signifying that all elements in the structure can be expressed as a combination of a limited number of generators.
The phrase is commonly encountered in academic and written contexts, particularly within mathematics, and can be less frequently found in casual oral speech. Its frequency of use is higher in specialized literature, lectures, and discussions among mathematicians.
"A finitely generated structure can help simplify complex problems in algebra."
"Una estructura finitamente generada puede ayudar a simplificar problemas complejos en álgebra."
"In group theory, a finitely generated structure is particularly important for understanding group properties."
"En la teoría de grupos, una estructura finitamente generada es particularmente importante para entender las propiedades de los grupos."
"Mathematicians often analyze finitely generated structures to explore their behavior."
"Los matemáticos a menudo analizan estructuras finitamente generadas para explorar su comportamiento."
While "finitely generated structure" is a specific term in mathematics and doesn't typically appear in idiomatic expressions, the concept of "finitely generated" can be related to various mathematical contexts. Here are a few related expressions:
Finitely generated group - A group that can be generated by a finite number of elements.
"A finitely generated group can be represented by a simple diagram."
"Un grupo finitamente generado puede ser representado por un diagrama simple."
Finitely generated module - A module over a ring that is generated by a finite set of elements.
"In algebra, a finitely generated module often leads to important results."
"En álgebra, un módulo finitamente generado a menudo conduce a resultados importantes."
Finitely generated abelian group - A specific type of group that is both finitely generated and abelian (commutative).
"Every finitely generated abelian group can be decomposed into a direct sum of cyclic groups."
"Cada grupo abeliano finitamente generado se puede descomponer en una suma directa de grupos cíclicos."
The term "finitely generated" comes from the combination of "finite," which derives from the Latin "finitus" meaning "limited" or "bounded," and "generate," which originates from the Latin "generare," meaning "to produce" or "to create." The word "structure" comes from the Latin "structura," indicating the arrangement or formation of components.
This comprehensive overview covers the aspects of "finitely generated structure" pertinent to mathematical language and its contextual use.