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For example, X 1 X 2 does not equal X 2 X 1. More generally, one can construct the free algebra R E on any set E of generators. Since rings may be regarded as Z-algebras, a free ring on E can be defined as the free algebra Z E . Over a field, the free algebra on n indeterminates can be constructed as the tensor algebra on an n-dimensional ...
Similarly, the one-point set is a generator for the category of sets. In fact, any nonempty set is a generator. In the category of sets, any set with at least two elements is a cogenerator. In the category of modules over a ring R, a generator in a finite direct sum with itself contains an isomorphic copy of R as a direct summand.
Free algebra. Clifford algebra • Geometric algebra Operator algebra. In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of ...
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set .
The Hasse diagram of the free Boolean algebra on two generators, p and q. Take p (left circle) to be "John is tall" and q (right circle)to be "Mary is rich". The atoms are the four elements in the row just above FALSE. The generators of a free Boolean algebra can represent independent propositions. Consider, for example, the propositions "John ...
Macaulay2 is built around fast implementations of algorithms useful for computation in commutative algebra and algebraic geometry. This core functionality includes arithmetic on rings, modules, and matrices, as well as algorithms for Gröbner bases, free resolutions, Hilbert series, determinants and Pfaffians, factoring, and similar.
In algebra, the free product of a family of associative algebras, over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the 's. The free product of two algebras A , B is denoted by A ∗ B .
Both binaries and source code are available for SageMath from the download page. If SageMath is built from source code, many of the included libraries such as OpenBLAS, FLINT, GAP (computer algebra system), and NTL will be tuned and optimized for that computer, taking into account the number of processors, the size of their caches, whether there is hardware support for SSE instructions, etc.