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In mathematics, '''BF algebras''' are a class of algebraic structures arising out of a symmetric "Yin Yang" concept for Bipolar Fuzzy logic, the ...year=2007|issue= 2|pages= 119–128|doi=10.2478/s12175-007-0003-x|doi-access=free}} ...

...ndation for the definition of [[invertible sheaf|invertible sheaves]] in [[algebraic geometry]]. ...is said to be invertible if it is locally a [[free module]] of [[rank of a free module|rank]] 1. In other words, <math> M_P\cong R_P </math> for all [[prim ...

...o <math>\mathbb{R}^{2n}</math> for some ''n'', but is not isomorphic as an algebraic variety to <math>\mathbb{C}^n</math>.<ref>{{citation ...: Proceedings of the Conference Interesting Algebraic Varieties Arising in Algebraic Transformation Group Theory Held at the Erwin Schrödinger Institute, Vienna ...

...is a [[module (mathematics)|module]] which is close to being [[free module|free]]. ...enerated module]] ''F'' over ''R'' such that <math> M \oplus F</math> is a free module. ...

== Algebraic expressions == [[File:Exp-tree-ex-12.svg|thumb|270px|right|Binary algebraic expression tree equivalent to ((5 + z) / -8) * (4 ^ 2)]] ...

...\sqrt{2}</math> is not algebraic if <math display="inline"> 1<a </math> is algebraic by the [[Gelfond–Schneider theorem]]. ...rs and the field of real algebraic numbers are [[elementarily equivalent]] structures. ...

...roperties. Those classes can be defined in two distinct ways, using either algebraic notions or topological notions. Varieties of finite [[monoid]]s, varieties === Algebraic definition === ...

...=52|pages=161–199|doi=10.7146/math.scand.a-12001|issn=1903-1807|doi-access=free}}</ref> ...)</math></blockquote>An element of <math>x \in F(X)</math> is called an '''algebraic element''' if it is the henselization of an <math>\mathcal{O}_S</math>-alge ...

...0.1007/3-540-12896-4_359 |isbn=978-3-540-38775-6|hdl=1813/6405 |hdl-access=free }}</ref> Quotient types have been studied in the context of [[Martin-Löf ty ...g set]] is a quotient are also termed quotients. Examples of such quotient structures include quotient [[quotient set|sets]], [[quotient group|groups]], [[quotie ...

{{For|the song|Free Loop (One Night Stand)}} ...d point. Formally, let <math>X</math> be a [[topological space]]. Then a free loop in <math>X</math> is an [[equivalence class]] of [[continuous function ...

{{Short description|Algebraic structure}} ...characteristic of a field|characteristic]] not equal to 2, the theory of J-structures is essentially the same as that of Jordan algebras. ...

...rmation]]. Often, this revolves around using [[Category theory|categorical structures]] or [[Cohomology|cohomology theories]] to classify and describe various to ...e objects the categories describe some sort of [[quasiparticles]], and the structures on the category encode the ways in which these quasiparticles can interact ...

...S |last2=Vaught |first2=R |title=The first order properties of products of algebraic systems |journal=[[Fundamenta Mathematicae]] |date=1959 |volume=47 |issue=1 ...le=S. Feferman and R. L. Vaught. The first order properties of products of algebraic systems. Fundamenta mathematicae, vol, 47 (1959), pp. 57–103. (Review) |jou ...

By definition, <math>F_n</math> is a [[Quotient ring|quotient]] of the [[free ring]] <math>k\langle t_1, \dots, t_m \rangle</math> with <math>t_i \mapsto ...space in the [[noncommutative algebraic geometry]] (i.e., we don't demand free variables to commute) and thus a generic matrix ring of size ''n'' is the c ...

...ers in ''M''. This is analogous to the [[strongly minimal theory|minimal]] structures, which are exactly the analogous property down to equality. O-minimal structures can be defined without recourse to model theory. Here we define a structur ...

...their associated [[algebra (ring theory)|algebras]] are of much use in the algebraic study of these geometric objects. * Affine monoids are also '''[[Affine connection|torsion free]]'''. For an affine monoid <math>M</math>, <math>nx = ny</math> implies tha ...

{{Short description|Element of algebraic structure}} In [[mathematics]], a '''Garside element''' is an element of an [[algebraic structure]] such as a [[monoid]] that has several desirable properties. ...

...erated]] [[weak Hausdorff space]]s,''' '''CGWH,''' is a category used in [[algebraic topology]] as an alternative to the [[category of topological spaces]], ''' ...ath>\operatorname{Map}(X, Y)</math> or <math>Y^X</math> and is called the (free) '''mapping space''' from ''X'' to ''Y''. Moreover, there is a homeomorphi ...

...the study of algebraic structures on the [[homology theory|homology]] of [[free loop space]]s. The field was started by {{harvs|txt|first1=Moira|last1=Chas ...en through the second one. There is no analogous product structure for the free loop space <math>LX</math> of all maps from <math>S^1</math> to <math>X</ma ...

{{short description|Counterexample in algebraic geometry}} In [[algebraic geometry]], '''Hironaka's example''' is a non-Kähler [[complex manifold]] t ...