LITERATURE CITED i. H. Weyl, The Classical Groups, Their Invariants and Representations [Russian translation], IL, Moscow (1947). 2. V.S. Drenski, "Representations of the Symmetric group and manifolds of linear alge- bras," Mat. Sb., 115, 98-115 (1981). 3. A. R. Kemer, "Nonmatrix manifolds," Algebra Logika, 19. No. 3, 255-283 (1980). 4. A.P. Popov, "Identities of tensor product of two copies of Grassman algebra," Dokl. Bolg. Akad. Nauk, 34, No. 9, 1205-1208 (1981). STRUCTURE OF POWERS OF GENERALIZED INDEX SETS V. L. Selivanov UDC 517.11:518.5 In this article we generalize the results of [1-3] in order to relate problems concern- ing index sets more closely with the theory of complete numerations [4] and treat ordinary multiple reducibility from a single viewpoint. Let A = (A,~) be a numerated set and let the set S be arbitrary. The set M~(A,S) of all maps from A into S has two naturalorderings, which we denote by ~ and gM in order to emphasize their relationship with the corresponding concepts in [i]. Specifically, for ~,~E~ I~,S) we set ~ if ~o~o~ (here o denote compositionofmaps and ~ denotesredu- cibilityofnumerations), and ~M~ if ~=~o~ for some morphism ~ from 4 into A ~M~ implies that ~ . Among other things, we will study the preorders ~m and ~M We note that we recover the case of "ordinary" index sets by taking S={0,/} , in which case we identify ~(A,$) with the family of all subsets of A and ~0~ with the index set °¢-l({aEA I~(QI=~}). We will use some of the terminology in [i]. i. AUXILIAKY CONCEPTS We introduce some concepts needed to study the preorders ~m,~ If (~; ~) is a _ F~ X - - - ' " P preordered set, then the closure of a set X c p in (P; ~) is the set [XJ~-~E l (XE ]. Let f be maps from n into a preordered set then is equiva- lent to @ if Two preordered sets CP; > and are equivalent if there exist monotone maps ~:P~P~ ~t: p,_~p whose composite ~'o is equivalent to the identity map of ID; and ~o ~, equivalent to the identity map of ~' Let ~be a nonempty set. By a discrete generalized semilattice (more precisely, an ~ - discrete semilattice) we mean any algebraic system (~;~, I~}~E f) satisfying the following conditions: i) ~ is a preorder on ~ ; 2) ~Ef I~ ~) ; 3) for all ~,~ EF, ~b, the proposition ~LT~ (%£&A~ZA~Z A (ff6/2~A~ A~--~)A(~E~A~--~ ~ ~V~=~ is valid in ~. The element % , whose existence is asserted in 3), is defined uniquely up to equivalence in (~; ~ ) , so that we can define binary operations ~(6E~) on ~ (~(~,~)~f) such that: Translated from Algebra i Logika, Vol. 21, No. 4, 472-491, July-August, 1982. Original article submitted March 31, 1981. 316 0002-5232/82/2104 0316507.50 @ I¢~83 Plenum Publishing Corporation