The tensor product of two modules A and B over a commutative ring R is defined in exactly the same way as the tensor product of vector spaces over a field: ⊗:= (×) / where now F(A × B) is the free R-module generated by the cartesian product and G is the R-module generated by the same relations as above.. More generally, the tensor product can be defined even if the ring is non-commutative. Example 6.16 is the tensor product of the filter {1/4,1/2,1/4} with itself. This is a distributive product, and to make quotients of distributive products we need ideals. For other varieties, such as the variety of semigroups, the tensor product has been investigated more recently (5).

THE SEMILATTICE TENSOR PRODUCT OF DISTRIBUTIVE LATTICES BY GRANT A. FRASER ABSTRACT. Vertically we take the tensor product of the $\mathbb{Z}$’s creating.

The tensor product of groups and of rings have been studied extensively. In this paper we investigate the tensor product of distributive lattices. We define the tensor product A ® S for arbitrary semilat-tices A and B. The tensor product V ⊗ W is thus defined to be the vector space whose elements are (complex) linear combinations of elements of the form v ⊗ w, with v ∈ V,w ∈ W, with the above rules for manipulation. 1 + v ⊗w. In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g.

Ideals are subspaces that absorb products on the right, such as. The construction is analogous to one used in ring theory (see 14], [7], [8]) and different from one studied by A. Waterman [12], D. Mowat [9], and Z. Shmuely [10]. While we have seen that the computational molecules from Chapter 1 can be written as tensor products, not all computational molecules can be written as tensor products: we need of course that the molecule is a rank 1 matrix, since matrices which can be written as a tensor product always have rank 1. The well-known algebraic concept of tensor product exists for any variety of algebras. We thus demand distributive properties for the tensor product: (v1 + v2)⊗w = v1 ⊗w + v2 ⊗w, (1.3) v ⊗(w. 1 + w. 2) = v ⊗w. left ideals absorb products on the left. So … 2.