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

2 STEVEN DUPLIJ bijections by ϵ . On the Cartesian product G × n one can define a polyadic ( n -ary, n -adic, if it is necessary to specify n , its arity or rank) operation µ n : G × n → G . For operations we use small Greek letters and place arguments in square brackets µ n [ g ] . The operations with n = 1 , 2 , 3 are called unary, binary and ternary . The case n = 0 is special and corresponds to fixing a distinguished element of G , a “constant” c ∈ G , and it is called a 0-ary operation µ ( c ) 0 , which maps the one-point set {•} to G , such that µ ( c ) 0 : {•} → G , and (formally) has the value µ ( c ) 0 [ {•} ] = c ∈ G . The composition of n -ary and m -ary operations µ n ◦ µ m gives ( n + m − 1) -ary operation by the iteration µ n + m − 1 [ g , h ] = µ n [ g , µ m [ h ]] . If we compose µ n with the 0-ary operation µ ( c ) 0 , then we obtain the arity “collapsing” µ ( c ) n − 1 [ g ] = µ n [ g , c ] , because g is a polyad of length ( n − 1) . Definition 2.1. A polyadic system G = ⟨ set | operations ⟩ is a set G which is closed under polyadic operations. If a concrete polyadic system 1 has one fundamental n -ary operation, it is called polyadic multiplication (or n -ary multiplication ) µ n . A n -ary system G n = ⟨ G | µ n ⟩ is a set G closed under one n -ary operation µ n (without any other additional structure). For a given n -ary system ⟨ G | µ n ⟩ one can construct another polyadic system ⟨ G | µ ′ n ′ ⟩ over the same set G , but with another multiplication µ ′ n ′ of different arity n ′ . In general, there are three ways of changing the arity: (1) Iterating . Composition of the operation µ n with itself increases the arity from n to n ′ = n iter > n . We denote the number of iterating multiplications by ℓ µ and call the resulting composition an iterated product 2 µ ℓ µ n (using the bold Greek letters) as (or µ • n if ℓ µ is obvious or not important) µ ′ n ′ = µ ℓ µ n def = ℓ µ z }| { µ n ◦ ( µ n ◦ . . . ( µ n × id × ( n − 1) ) . . . × id × ( n − 1) ) , (2.1) where the final arity is n ′ = n iter = ℓ µ ( n − 1) + 1 . (2.2) There are many variants of placing µ n ’s among id ’s in the r.h.s. of (2.1), if no associativity is assumed. An example of the iterated product can be given for a ternary operation µ 3 ( n = 3 ), where we can construct a 7-ary operation ( n ′ = 7 ) by ℓ µ = 3 compositions µ ′ 7 [ g 1 , . . . , g 7 ] = µ 3 3 [ g 1 , . . . , g 7 ] = µ 3 [ µ 3 [ µ 3 [ g 1 , g 2 , g 3 ] , g 4 , g 5 ] , g 6 , g 7 ] , (2.3) and the corresponding commutative diagram is G × 7 µ 3 × id × 4 - G × 5 µ 3 × id × 2 - G × 3 HHHHHHHHHHH µ ′ 7 = µ 3 3 j G µ 3 ? (2.4) In the general case, the horizontal part of the (iterating) diagram (2.4) consists of ℓ µ terms. (2) Reducing (Collapsing) . To decrease arity from n to n ′ = n red < n one can use n c distinguished elements (“constants”) as additional 0 -ary operations µ ( c i ) 0 , i = 1 , . . . n c , such that 3 the reduced product is defined by µ ′ n ′ = µ ( c 1 ...c n c ) n ′ def = µ n ◦   n c z }| { µ ( c 1 ) 0 × . . . × µ ( c n c ) 0 × id × ( n − n c )   , (2.5) where n ′ = n red = n − n c , (2.6) 1 A set with one closed binary operation without any other relations was called a groupoid by Hausmann and Ore H AUSMANN AND O RE [1937] (see, also C LIFFORD AND P RESTON [1961]). Nowadays the term “groupoid” is widely used in the category theory and homotopy theory for a different construction, the so-called Brandt groupoid B RANDT [1927]. Bourbaki B OURBAKI [1998] introduced the term “magma”. To avoid misreading we take some neutral notations “polyadic system” and “ n -ary system”. 2 Sometimes µ ℓ µ n is named a long product D ¨ ORNTE [1929]. 3 In D UDEK AND M ICHALSKI [1984] µ ( c 1 ...c nc ) n is called a retract, which is already a busy and widely used term in category theory for another construction.