A "q-DEFORMED" GENERALIZATION OF THE HOSSZU-GLUSKIN THEOREM

A “q-DEFORMED” GENERALIZATION OF THE HOSSZ ´U-GLUSKIN THEOREM STEVEN DUPLIJ A BSTRACT . In this paper a new form of the standard Hossz´u-Gluskin theorem in terms of polyadic powers and in the language of diagrams is presented. It is shown that the Hossz´u-Gluskin chain formula is not unique and can be generalized (“deformed”) using a parameter q which takes special integer values. The invariance version of the “q- deformed” analog of the Hossz´u-Gluskin theorem is formulated, and some examples are considered. A “q-deformed” homomorphism theorem is also given. C ONTENTS 1. Introduction 1 2. Preliminaries 1 3. Standard Hossz´u-Gluskin theorem 7 4. “Deformation” of Hossz´u-Gluskin chain formula 12 5. Generalized “deformed” version of the homomorphism theorem 16 References 17 1. I NTRODUCTION Starting from the early days of “polyadic history” K ASNER [1904], P R ¨ UFER [1924], D ¨ ORNTE [1929], the interconnection between polyadic systems and binary ones has been one of the main areas of interest P OST [1940], T HURSTON [1954]. First constructions were confined to building some special polyadic (mostly ternary C ERTAINE [1943], L EHMER [1932]) operations on elements of binary groups M ILLER [1935], L EHMER [1936], C HUNIKHIN [1946]. The very special form of n -ary multiplication in terms of binary multiplication and a special mapping as a chain formula was found in H OSSZ ´ U [1963] and G LUSKIN [1964, 1965]. A theorem which claims that any n -ary multiplication can be presented in this form is called a Hossz´u-Gluskin theorem (for review see D UDEK AND G ŁAZEK [2008], G AL ’ MAK [2003]). A concise and clear proof of the Hossz´u-Gluskin chain formula was presented in S OKOLOV [1976]. In this paper we give a new form of the standard Hossz´u-Gluskin theorem in terms of polyadic powers. Then we show that the Hossz´u-Gluskin chain formula is not unique and can be generalized (“deformed”) using a parameter q which takes special integer values. We present the invariance version of the “ q -deformed” analog of the Hossz´u- Gluskin theorem and consider examples. A “ q -deformed” homomorphism theorem is also given. 2. P RELIMINARIES We will use the concise notations from our previous review paper D UPLIJ [2012], while here we repeat some necessary definitions using the language of diagrams. For a non-empty set G , we denote its elements by lower-case Latin letters g i ∈ G and the n -tuple (or polyad ) g 1 , . . . , g n will be written by ( g 1 , . . . , g n ) or using one bold letter with index g ( n ) , and an n -tuple with equal elements by g n . In case the number of elements in the n -tuple is clear from the context or is not important, we denote it in one bold letter g without indices. We omit g ∈ G , if it is obvious from the context. The Cartesian product n z }| { G × . . . × G = G × n consists of all n -tuples ( g 1 , . . . , g n ) , such that g i ∈ G , i = 1 , . . . , n . The i -projection of the Cartesian product G n on its i -th “axis” is the map Pr ( n ) i : G × n → G such that ( g 1 , . . . g i , . . . , g n ) 7 −→ g i . The i -diagonal Diag n : G → G × n sends one element to the equal element n -tuple g 7 −→ ( g n ) . The one-point set {•} is treated as a unit for the Cartesian product, since there are bijections between G with G × {•} × n , where G can be on any place. In diagrams, if the place is unimportant, we denote such 2010 Mathematics Subject Classification. 08B05, 17A42, 20N15. 1