Commutative monads as a theory of distributions
... T -algebras deserve the name T -linear spaces, and homomorphisms deserve the name T linear maps (and, if T is understood from the context, the ‘T ’ may even be omitted). This allows us to talk about partial T -linear maps, as well as T -bilinear maps, as we shall explain. An example of a commutative ...
... T -algebras deserve the name T -linear spaces, and homomorphisms deserve the name T linear maps (and, if T is understood from the context, the ‘T ’ may even be omitted). This allows us to talk about partial T -linear maps, as well as T -bilinear maps, as we shall explain. An example of a commutative ...
© Sherry Scarborough, Lynnette Cardenas 7/8/2005 ... polygon is the sum of the lengths of the sides... Math 366 Study Guide (revised with thanks to Lynnette Cardenas)
... regions by drawing in one diagonal, but the two triangular regions are not necessarily congruent. Although the bases of the two triangular regions may be different, the height of both - since each is part of the same trapezoidal region - will be the same. If the base of the top triangle is b1 and th ...
... regions by drawing in one diagonal, but the two triangular regions are not necessarily congruent. Although the bases of the two triangular regions may be different, the height of both - since each is part of the same trapezoidal region - will be the same. If the base of the top triangle is b1 and th ...
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Group action
In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A group action formalizes of the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices. It is the same as a group action of G on an ordered basis of a vector space.A group action is an extension to the notion of a symmetry group in which every element of the group ""acts"" like a bijective transformation (or ""symmetry"") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.If G is a group and X is a set, then a group action may be defined as a group homomorphism h from G to the symmetric group on X. The action assigns a permutation of X to each element of the group in such a way that the permutation of X assigned to the identity element of G is the identity transformation of X; a product gk of two elements of G is the composition of the permutations assigned to g and k.The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Despite this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.