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Free product. In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “ universal ” group having these properties, in the sense that any two ...
In algebra, the free product ( coproduct) 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. The notion is a ring-theoretic analog of a free product of groups .
proprietary software, such as Microsoft Office, Windows, Adobe Photoshop, Facebook or FaceTime. Users cannot study, change, and share their source code. freeware or gratis [ 14] software, which is a category of proprietary software that does not require payment for basic use.
Direct products The direct product of a family of R-algebras is the ring-theoretic direct product. This becomes an R-algebra with the obvious scalar multiplication. Free products One can form a free product of R-algebras in a manner similar to the free product of groups. The free product is the coproduct in the category of R-algebras. Tensor ...
The free group FS is defined to be the group of all reduced words in S, with concatenation of words (followed by reduction if necessary) as group operation. The identity is the empty word. A reduced word is called cyclically reduced if its first and last letter are not inverse to each other.
Freemium, a portmanteau of the words "free" and "premium", is a pricing strategy by which a basic product or service is provided free of charge, but money (a premium) is charged for additional features, services, or virtual (online) or physical (offline) goods that expand the functionality of the free version of the software. [ 1][ 2] This ...
"Free and open-source software" (FOSS) is an umbrella term for software that is simultaneously considered both free software and open-source software. [5] The precise definition of the terms "free software" and "open-source software" applies them to any software distributed under terms that allow users to use, modify, and redistribute said software in any manner they see fit, without requiring ...
Some freeware products are released alongside paid versions that either have more features or less restrictive licensing terms. This approach is known as freemium ("free" + "premium"), since the free version is intended as a promotion for the premium version. [17] The two often share a code base, using a compiler flag to determine which is ...
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