Mathematics & Physics 2015, 8(1), 49–54 УДК 517.9 The Tensor Product and Quasiorder of an Algebra Related to Cohen-Macaulay Rings Ali Molkhasi∗ Department of Mathematics University of Tabriz Tabriz Iran Received 01.10.2014, received in revised form 10.11.2014, accepted 10.12.2014 This paper shows how the tensor products of the distributive lattices and the finite solvable groups can used to WB-height-unmixed of the method of Stanley and Reisner. <...> Introduction N.Funayama and T.Nakayama proves that congruence relations on an arbitrary lattice have an interesting connection with distributive lattices. <...> For terminology and basic results of lattice theory and universal algebra see [3, 8], and [9]. <...> By using distributive lattice and Stanley-Reisner theory, we formulate new characterizations of Cohen-Macaulay ring. <...> Recall that the local ring R is Cohen-Macaulay when so is R as an R-module. <...> A Noetherian ring (which may not be local) R is said to be Cohen-Macaulay when its localization at any maximal ideal is Cohen-Macaulay local. <...> In this paper relationships among quasiorder of an algebra, Cohen-Macaulay rings, the order complex of the lattice of all subgroups of a finite group and polytopes are considered. <...> We denote the set of all quasiorders of an algebra A = (A,F), the set of all almost principal ideals of the lattice distributive L, the set of all almost principal filters of the lattice distributive L, and the lattice of all congruence relations of lattice L by Quord(A), I(L), ̥(L), and Con(L), respectively. <...> Notice that Rd is the d-dimensional Euclidean space, S ⊆ Rd is a polytope and K is a ring, and E = K[S]. <...> Also, if A is an algebra in any n-permutable variety, then E[Con(A)][X1, X2, . . .] is WB-height-unmixed. <...> Finally in section 2, it is shown that if G is a finite solvable group and ℜ is the order complex of L(G), the set of all subgroups of G, then ℜ[X1, X2, . . .] is WB-height-unmixed. <...> Finally, it is prove that if C and B are distributive lattices, then E[C⊗B][X1, X2, . . .] isWB-height-unmixed. 1. <...> Quasiorder of an algebra and the polytopes In this section relationship among quasiorder of an Algebra, the Cohen-Macaulay ring, and the polytopes are considered. <...> Let we start the detailed <...>