aa r X i v : . [ m a t h . G R ] F e b ON PROFINITE POLYADIC GROUPS
M. SHAHRYARI AND M. ROSTAMIAbstract.
We study the structure of profinite polyadic groups andwe prove that a polyadic topological group (
G, f ) is profinite, if andonly if, it is compact, Hausdorff, totally disconnected, and for any opencongruence R ⊆ G × G , the quotient polyadic group ( G/R, f R ) is finite. Introduction
In this article, we study the structure of profinite polyadic groups: polyadicgroups which are the inverse limit of a system of finite polyadic groups. Apolyadic group is a natural generalization of the concept of group to thecase where the binary operation of group replaced with an n -ary associa-tive operation, one variable linear equations in which have unique solutions.So, in this article, polyadic group means an n -ary group for a fixed naturalnumber n ≥
2. These interesting algebraic objects are introduced by Kas-ner and D¨ornte ([8] and [2]) and studied extensively by Emil Post duringthe first decades of the last century, [12]. During decades, many articles arepublished on the structure of polyadic groups. Already homomorphisms andautomorphisms of polyadic groups are studied in [10]. A characterization ofthe simple polyadic groups is obtained by them in [11]. Also, the represen-tation theory of polyadic groups is studied in [6] and the complex charactersof finite polyadic groups are also investigated in [13]. The structure of freepolyadic groups is determined in [1], [ ? ], and [9].It is easy to define topological polyadic groups, and so, one can ask whichtopological polyadic groups are profinite. In this paper, we study this prob-lem and as the main result, we prove that a polyadic topological group ( G, f )is profinite, if and only if, it is compact, Hausdorff, totally disconnected, andfor any open congruence R ⊆ G × G , the quotient polyadic group ( G/R, f R )is finite. 2. Polyadic groups
A polyadic group is a pair (
G, f ) where G is a non-empty set and f : G n → G is an n -ary operation, such that Date : February 2, 2021.
MSC(2010): 20N15Keywords: Polyadic groups; n -ary groups; Profinite groups and polyadic groups; Post’s cover andretract of a polyadic group . M. SHAHRYARI AND M. ROSTAMI i- the operation is associative, i.e. f ( x i − , f ( x n + i − i ) , x n − n + i ) = f ( x j − , f ( x n + j − j ) , x n − n + j )for any 1 ≤ i < j ≤ n and for all x , . . . , x n − ∈ G , andii- for all a , . . . , a n , b ∈ G and 1 ≤ i ≤ n , there exists a unique element x ∈ G such that f ( a i − , x, a ni +1 ) = b. Note that, here we use the compact notation x ji for every sequence x i , x i +1 , . . . , x j of elements in G , and in the special case when all terms of this sequence areequal to a fixed x , we denote it by ( t ) x , where t is the number of terms.Clearly, the case n = 2 is exactly the definition of ordinary groups. Duringthis article, we assume that n is fixed. Note that an n -ary system ( G, f ) ofthe form f ( x n ) = x x . . . x n b , where ( G, · ) is a group and b a fixed elementbelonging to the center of ( G, · ), is a polyadic group, which is called b -derived from the group ( G, · ) and it is denoted by der nb ( G, · ). In the case when b is the identity of ( G, · ), we say that such a polyadic group is reduced to thegroup ( G, · ) or derived from ( G, · ) and we use the notation der n ( G, · ) for it.For every n >
2, there are n -ary groups which are not derived from anygroup. A polyadic group ( G, f ) is derived from some group if and only if, itcontains an element a (called an n -ary identity ) such that f ( ( i − a , x, ( n − i ) a ) = x holds for all x ∈ G and for all i = 1 , . . . , n , see [3].From the definition of an n -ary group ( G, f ), we can directly see that forevery x ∈ G , there exists only one y ∈ G , satisfying the equation f ( ( n − x , y ) = x. This element is called skew to x and it is denoted by x . As D¨ornte [2] proved,the following identities hold for all x, y ∈ G , 2 ≤ i ≤ n , f ( ( i − x , x, ( n − i ) x , y ) = f ( y, ( n − i ) x , x, ( i − x ) = y. These identities together with the associativity identities, axiomatize thevariety of polyadic groups in the algebraic language ( f, − ).Suppose ( G, f ) is a polyadic group and a ∈ G is a fixed element. Definea binary operation x ∗ y = f ( x, ( n − a , y ) . Then ( G, ∗ ) is an ordinary group, called the retract of ( G, f ) over a . Sucha retract will be denoted by ret a ( G, f ). All retracts of a polyadic group are
N PROFINITE POLYADIC GROUPS 3 isomorphic [5]. The identity of the group ( G, ∗ ) is a . One can verify thatthe inverse element to x has the form y = f ( a, ( n − x , x, a ) . One of the most fundamental theorems of polyadic group is the following,now known as
Hossz´u -Gloskin’s theorem . We will use it frequently in thisarticle and the reader can use [4], [ ? ], [7] and [14] for detailed discussions. Theorem 2.1.
Let ( G, f ) be a polyadic group. Then there exists an ordi-nary group ( G, • ) , an automorphism θ of ( G, • ) and an element b ∈ G suchthat1. θ ( b ) = b ,2. θ n − ( x ) = bxb − , for every x ∈ G ,3. f ( x n ) = x θ ( x ) θ ( x ) · · · θ n − ( x n ) b , for all x , . . . , x n ∈ G . According to this theorem, we use the notation der θ,b ( G, • ) for ( G, f ) andwe say that (
G, f ) is ( θ, b )-derived from the group ( G, • ).There is one more important ordinary group associated to a polyadicgroup which we call it the Post’s cover . This is the first fundamental theoremconcerning polyadic groups. The proof can be find in [12].
Theorem 2.2.
Let ( G, f ) be a polyadic group. Then, there exists a uniquegroup ( G ∗ , ◦ ) such that1- G is contained in G ∗ as a coset of some normal subgroup K .2- K is isomorphic to a retract of ( G, f ) .3- We have G ∗ /K ∼ = Z n − .4- Inside G ∗ , for all x , . . . , x n ∈ G , we have f ( x n ) = x ◦ x ◦ · · · ◦ x n .5- G ∗ is generated by G . The group G ∗ is also universal in the class of all groups having properties1, 4. More precisely, if β : ( G, f ) → der n ( H, ∗ ) is a polyadic homomorphism,then there exists a unique ordinary homomorphism h : G ∗ → H , such that h | G = β . This universal property characterizes G ∗ uniquely. The explicitconstruction of the Post’s cover can be find in [13].Finally, we have to mention that the structure of polyadic homomor-phisms will be needed in what follows. The reader can see [10] for details. Theorem 2.3.
Suppose ( G, f ) = der θ,b ( G, · ) and ( H, h ) = der η,c ( H, ∗ ) aretwo polyadic groups. Let ψ : ( G, f ) → ( H, h ) be a homomorphism. Thenthere exists a ∈ H and an ordinary homomorphism φ : ( G, · ) → ( H, ∗ ) , such M. SHAHRYARI AND M. ROSTAMI that ψ = R ( a ) φ , where R ( a ) denotes the map x x ∗ a . Further a and φ satisfy the following conditions; h ( ( n ) a ) = φ ( b ) ∗ a and φθ = I a ηφ, where, I a denotes the inner automorphism x a ∗ x ∗ a − . Conversely, if a and φ satisfy the above two conditions, then ψ = R a φ is a homomorphism ( G, f ) → ( H, h ) . Profinite polyadic groups
A profinite polyadic group is the inverse limit of an inverse system of finitepolyadic groups. More precisely, let ( I, ≤ ) be a directed set and suppose { ( G i , f i ) , ϕ ij , I } is an inverse system of finite polyadic groups. This meansthat for every pair ( i, j ) of elements of I with j ≤ i , we are given a polyadichomomorphism ϕ ij : ( G i , f i ) → ( G j , f j )such that the equality ϕ jk ϕ ij = ϕ ik holds for all k ≤ j ≤ i . Now, assumethat ( G, f ) = lim ←− i ( G i , f i ) . Then (
G, f ) is called a profinite polyadic group. Note that, for (
G, f ) to bea profinite group, it requires that G is non-empty, and we will see that itis indeed so. From now on, we consider the pair ( G, f ) which is the abovementioned inverse limit. A realization of this pair can be given as follows:Let Q i ( G i , f i ) be the direct product of the family { ( G i , f ) i ) } i ∈ I . This is apolyadic group with the n -ary operation( Y f i )(( x i ) , ( x i ) , . . . , ( x in )) = ( f i ( x i , x i , . . . , x in )) i ∈ I . Here of course, we denoted an arbitrary element of the direct product assequence ( a i ) i ∈ I or simply ( a i ). Now, we have G = { ( x i ) i ∈ I : ∀ j ≤ i ϕ ij ( x i ) = x j } , and hence f (( x i ) , ( x i ) , . . . , ( x in )) = ( f i ( x i , x i , . . . , x in )) i ∈ I . This realization allows us to consider the natural projection maps ϕ j : G → G i defined by ϕ j (( x i ) i ∈ I ) = x j , which are obviously polyadic homomorphisms.Note that, as each G i is finite, being a closed subspace of the directproduct of a family of finite sets, ( G, f ) is compact, Hausdorf, and totallydisconnected topological polyadic group, of course, if it has been shown that G is non-empty. Indeed, using standard topological arguments, we can provethat G = ∅ as every G i is compact. N PROFINITE POLYADIC GROUPS 5
Recall that, according to Hossz´u -Gloskin’s theorem, we have ( G i , f i ) =der θ i ,b i ( G i , • i ), for some ordinary group ( G i , • i ), an element b i ∈ G i , and anautomorphism θ i , satisfying the conclusions of Theorem 2.1. We will provethat in some sense, there exists a binary operation • on G such that( G, • ) = lim ←− i ( G i , • i ) , and hence ( G, • ) will be proved to be profinite. Consider the polyadic homo-morphism ϕ ij . According to Theorem 2.3, there exist an element a ij ∈ G j ,and a group homomorphism ψ ij : ( G i , • i ) → ( G j , • j ), such that ϕ ij = R ( a ij ) ψ ij . Further, we have the following equalities:1 . f j ( a ij , a ij , . . . , a ij ) = ψ ij ( b i ) • j a ij ,2 . ψ ij θ i = I ( a − ij ) θ j ψ ij .For any triple of indices k ≤ j ≤ i , we have ϕ ij = R ( a ij ) ψ ij , ϕ ik = R ( a ik ) ψ ik , ϕ jk = R ( a jk ) ψ jk , therefore a ij = ϕ ij (1) , a ik = ϕ ik (1) , a jk = ϕ jk (1) . Note that in each equality, 1 is the identity element of the correspondinggroup. Since ϕ ik = ϕ jk ϕ ij , so we have a ik = ϕ jk ( a ij ) . Now, let Y i be the set of all sequences ( x j ) (in the direct product) such thatfor any j and k ≤ i , we have ϕ jk ( x j ) = x k . This set is non-empty, becausewe can consider a sequence where x j = a ij , for j ≤ i , and for all other j , x j is arbitrary. This sequence will be an element of Y i . The set Y i is closedand if i ≤ s , then Y s ⊆ Y i . As the direct product is compact, and the family { Y i } has finite intersection property, we have \ i Y i = ∅ , showing that G is not empty.Now, we can prove our first main result: Theorem 3.1.
Let ( G, f ) = der θ,b ( G, • ) be a profinite polyadic group. Thenthe ordinary group ( G, • ) is profinite.Proof. Let (
G, f ) be the inverse limit of the inverse system { ( G i , f i ) , ϕ ij , I } ,where every ( G i , f i ) is a finite polyadic group. As G = ∅ , we choose anarbitrary element ( v i ) ∈ G . We know that all retracts of a polyadic groupare isomorphic to each other. So, we consider the retract( G i , • i ) = ret v i ( G i , f i ) . M. SHAHRYARI AND M. ROSTAMI
By the construction of Sokolov (see [14]), we have θ i ( x ) = f i ( v i , x, ( n − v i ) , for any x . Also we have b i = f i ( v i , . . . , v i ) . Using this special form of the retract, we see that the maps ϕ ij are grouphomomorphisms as well, because ϕ ij ( x • i y ) = ϕ ij ( f i ( x, ( n − v i , y ))= f j ( ϕ ij ( x ) , ( n − ϕ ij ( v i ) , ϕ ij ( y ))= f j ( ϕ ij ( x ) , ( n − v j , ϕ ij ( y ))= ϕ ij ( x ) • j ϕ ij ( y ) . Note that, here we use the fact ϕ ij ( v i ) = v j as we assumed that ( v i ) ∈ G .This shows that the maps ϕ ij : G i → G j are in the same time, grouphomomorphisms and { ( G i , • i ) , ϕ ij , I } is an inverse system of finite groups.Obviously, ( G, • ) is the inverse limit of this system and so it is a profinitegroup. (cid:3) Note that in some sense, the inverse of the above theorem is also true: if weconsider a profinite group ( G, • ) together with a continuous automorphism θ and an element b satisfying the requirement of 2.1, then the polyadic groupder θ,b ( G, • ) will be profinite. One may ask also about the automorphism θ in the above proof. The above construction shows that, for any ( x i ) ∈ G ,we have θ (( x i ) i ∈ I ) = ( θ i ( x i )) i ∈ I . As a result of the above theorem, we can see that the Post’s cover of aprofinite polyadic group is also profinite.
Corollary 3.2.
Let ( G, f ) be profinite and G ∗ be the Post’s cover. Then G ∗ is also profinite.Proof. We know that there exists a normal subgroup K of the Post’s coverwhich has the index n − K is isomorphic to the retract ( G, • ). Hence, K is profinite. Now, being a finite extension of a profinite group, G ∗ is alsoprofinite. (cid:3) We are ready now, to give a characterization of the profinite polyadicgroups.
Theorem 3.3.
A polyadic topological group ( G, f ) is profinite, if and only if,it is compact, Hausdorff, totally disconnected, and for any open congruence R ⊆ G × G , the quotient polyadic group ( G/R, f R ) is finite. N PROFINITE POLYADIC GROUPS 7
Proof.
We already saw that a profinite polyadic group is compact, Hausdorffand totally disconnected. Suppose R ⊆ G × G is an open congruence of thepolyadic group ( G, f ). In [11], it is proved that in this case, the equivalencerelation R is a subgroup of the ordinary group G × G (the letter G herestands for the group ( G, • )). This means that R is an open subgroup of theordinary profinite group G × G , and hence, the quotient ( G × G ) /R is finite.Let [ x ] R denotes the equivalence class of x ∈ G . So, we define a map ψ : GR → G × GR by ψ ([ x ] R ) = ( x, R . This map is well-defined as if we suppose [ x ] R = [ y ] R ,then ( x, y ) ∈ R and so ( x, y ) • R = R . This means that( x, R = (1 , y − ) R, and as in the quotient we have (1 , y − ) R = ( y, R , so the map is well-defined. Also, it is injective, since if ( x, • R = ( y, • R , then ( x − y, ∈ R ,so, we have also ( y, x ) = ( x, x ) • ( x − y, ∈ R. This shows that ψ is injective and hence the polyadic group G/R is finite.Conversely, suppose that the polyadic group (
G, f ) = der θ,b ( G, • ) is com-pact, Hausdorff, and totally disconnected. Further, suppose for any opencongruence R ⊆ G × G , the quotient polyadic group G/R is finite. We provethat the polyadic group is profinite. To do this, we consider the retract( G, • ). This group is compact, Hausdorff, and totally disconnected as well.So, we show that for any open normal subgroup K E ( G, • ), the group G/K is finite. This will show that the retract is profinite and hence, accordingto what we said already, the polyadic group will be profinite. Note that, ingeneral, there is no a simple correspondence between congruences of (
G, f )and normal subgroups of ( G, • ) (see [11]). For this reason, we argue directly:consider an open normal subgroup K E ( G, • ). Let R = n − [ i =0 θ i ( K × K ) . Obviously, R is a θ -invariant normal subgroup of G × G . Note that againhere we use G for the retract ( G, • ). This subgroup is open as K × K ⊆ R .Let R be the congruence generated by R . A simple argument shows that R is also an open subgroup of G × G (we will need this fact in what follows).By our assumption, the polyadic group G/R is finite. Define a map λ : n − [ i =0 Gθ i ( K ) → GR by λ ( xθ i ( K )) = [ x ] R . M. SHAHRYARI AND M. ROSTAMI
This map is well defined because if xθ i ( K ) = yθ i ( K ), then x − y ∈ θ i ( K )and hence ( x − y, ∈ θ i ( K × K ) ⊆ R . Since R is a subgroup of G × G , and( x, x ) ∈ R , so ( y, x ) = ( x, x )( x − y, ∈ R, which means [ x ] R = [ y ] R , proving that the map λ is well-defined. We showthat this map is n to 1. Suppose [ x ] R = [ y ] R . So ( x, y ) ∈ R and again wesee that ( x − y, ∈ R . This means that there are elements c , c , . . . , c s such that c = x − y , c s = 1, and for every 0 ≤ i ≤ s −
1, we have ( c i , c i +1 ) ∈ R . Starting from i = s −
1, we see that ( c s − , ∈ θ i ( K × K ), for some i .In other words, ( θ − i ( c s − ) , ∈ K × K . Similarly ( c s − , c s − ) ∈ θ i ( K × K ),for some i , and hence ( θ − i ( c s − ) , θ − i ( c s − )) ∈ K × K . This shows that( θ − i ( c s − ) , ∈ K × K and the similar statement is true for all smallerindices, especially, ( θ − i ( x − y ) , ∈ K × K , for some i . This means thatfor some r , we have x − y ∈ θ r ( K ), and hence xθ r ( K ) = yθ r ( K ). Therefore,the map λ is n to 1. This proves that G/K is finite and hence ( G, • ) isprofinite. (cid:3) References [1] Artamonov V.,
Free n -groups , Matematicheskie Zametki, 1970, , pp. 499-507.[2] D¨ornte W., Unterschungen ¨uber einen verallgemeinerten Gruppenbegriff , Math. Z.,1929, , pp. 1-19.[3] Dudek W., Remarks on n -groups , Demonstratio Math., 1980, , pp. 65-181.[4] Dudek W., Glazek K., Around the Hossz´u-Gluskin Theorem for n -ary groups , DiscreteMath., 2008, , pp. 4861-4876.[5] Dudek W. , Michalski J., On retract of polyadic groups , Demonstratio Math., 1984, , pp. 281-301.[6] Dudek W., Shahryari M., Representation theory of polyadic groups , Algebras andRepresentation Theory, 2012, , pp. 29-51.[7] Hossz´u M., On the explicit form of n -groups , Publ. Math., 1963, , pp. 88-92.[8] Kasner E., An extension of the group concept , Bull. Amer. Math. Soc., 1904, , pp.290-291.[9] Khodabandeh H., On free polyadic groups , 2019, arXiv:1905-1182.[10] Khodabandeh H., Shahryari M.,
On the representations and automorphisms ofpolyadic groups , Communications in Algebra, 2012, , pp. 2199-2212.[11] Khodabandeh H., Shahryari M., Simple polyadic groups , Siberian Math. Journal,2014, , pp. 734-744.[12] Post E., Polyadic groups , Trans. Amer. Math. Soc., 1940, , pp. 208-350.[13] Shahryari M., Representations of finite polyadic groups , Communications in Algebra,2012, , pp. 1625-1631.[14] Sokolov E., On the Gluskin-Hossz´u theorem for Dornte n -groups , Mat. Issled., 1976, , pp. 187-189.[15] Ribes L., Zalesskii P., Profinite groups, second edition , Springer, 2010.
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M. Shahryari, Department of Mathematics, College of Sciences, SultanQaboos University, Muscat, OmanM. Rostami, Department of Pure Mathematics, Faculty of MathematicalSciences, University of Tabriz, Tabriz, Iran
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