In the present paper: 1) an NP--polyagroup is defined as a
generalization of an $n-$group for $ngeq 3;$ and 2)
NP--polyagroups of the type $(s,n-1)$ is described as algebras
of the type $$ [$,=<kcdot s+1,kcdot s,kcdot
s-1>; k>1,sgeq 1$ ]

J. Ušan

R. Galić

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Mathematical Communications 6(2001), 153-159 153On NP - polyagroupsJanez Usˇan∗ and Radoslav Galic´†Abstract. In the present paper: 1) an NP–polyagroup is definedas a generalization of an n−group for n ≥ 3; and 2) NP–polyagroups ofthe type (s, n− 1) is described as algebras of the type < n, n− 1, n− 2 >[=< k · s+ 1, k · s, k · s− 1 >; k > 1, s ≥ 1 ].Key words: {1, n}-neutral operation, n-group, n-semigroup, n-qu-asigroup, Ps-associative n-groupoid, P–polyagroup, NP–polyagroupAMS subject classifications: 20N15Received September 21, 2001 Accepted November 14, 20011. PreliminariesDefinition 1. Let n ≥ 2 and let (Q,A) be an n-groupoid. We say that (Q,A) isa Do¨rnte n-group [ briefly: n-group ] iff it is an n-semigroup and an n-quasigroupas well.Remark 1. A notion of an n-group was introduced by W. Do¨rnte in [1] as ageneralization of the notion of a group. See, also [2–4].Proposition 1 [10]. Let n ≥ 2 and let (Q,A) be an n-groupoid. Then thefollowing statements are equivalent :(i) (Q,A) is an n-group;(ii) there are mappings −1and e of the sets Qn−1 and Qn−2, respectively, into theset Q such that the following laws hold in the algebra (Q, {A, −1, e}) [ of thetype 〈n, n− 1, n− 2〉 ](a) A(xn−21 , A(x2n−2n−1 ), x2n−1) = A(xn−11 , A(x2n−1n )),(b) A(e(an−21 ), an−21 , x) = x and(c) A((an−21 , a)−1, an−21 , a) = e(an−21 ); and∗Institute of Mathematics, University of Novi Sad, Trg D. Obradovic´a, 21 000 Novi Sad, Yu-goslavia, e-mail: jus@EUnet.yu†Faculty of Electrical Engineering, University of Osijek, Kneza Trpimira 2B, HR-31 000 Osijek,Croatia, e-mail: Galic.Radoslav@etfos.hr154 J.Usˇan and R.Galic´(iii) there are mappings −1and e of the sets Qn−1 and Qn−2, respectively, into theset Q such that the following laws hold in the algebra (Q, {A, −1, e}) [ of thetype 〈n, n− 1, n− 2〉 ](a¯) A(A(xn1 ), x2n−1n+1 ) = A(x1, A(xn+12 ), x2n−1n+2 ),(b¯) A(x, an−21 , e(an−21 )) = x(c¯) A(a, an−21 , (an−21 , a)−1) = e(an−21 ).Remark 2. e is a {1, n}-neutral operation of an n-groupoid (Q,A) iff algebra(Q, {A, e}) of the type 〈n, n−2〉 satisfies the laws (b) and (b¯) from Proposition 1 [: [7]]. The notion of an {i, j}-neutral operation (i, j ∈ {1, ..., n}, i < j) of an n-groupoidis defined in a similar way [: [7] ]. Every n-groupoid has at most one {i, j}-neutraloperation [: [7] ]. In every n-group (n ≥ 2) there is an {1, n}-neutral operation [: [7]]. There are n-groups without an {i, j}-neutral operation with {i, j} 	= {1, n} [ :[9]]. In [9], n-groups with {i, j}-neutral operations, for {i, j} 	= {1, n} are described.Operation −1from Proposition 1 [ (c), (c¯)] is a generalization of the inverse operationin a group. In fact, if (Q,A) is an n-group, n ≥ 2, then for every a ∈ Q and forevery sequence an−21 over Q(an−21 , a)−1def= E(an−21 , a, an−21 ),where E is a {1, 2n−1}-neutral operation of the (2n−1)-group (Q, 2A);2A (x2n−11 )def=A(A(xn1 ), x2n−1n+1 )[: [8] ]. (For n = 2, a−1 = E(a); a−1 is the inverse element ofelement a with respect to the neutral element e(∅) of the group (Q,A).)Definition 2. Let k > 1, s ≥ 1, n = k · s + 1 and let (Q,A) be an n−groupoid.Then, we say that (Q,A) is a partiallys−associative (briefly: Ps–associative)n−groupoid iff for every i, j ∈ {t · s+ 1|t ∈ {0, 1, . . . , k}}, i < j, the following lawholdsA(xi−11 , A(xi+n−1i ), x2n−1i+n ) = A(xj−11 , A(xj+n−1j ), x2n−1j+n )[: < i, j > −associative law ].Remark 3. For s = 1 (Q,A) is a (k + 1)−semigroup; k > 1. A notion of ans-associative n-groupoid was introduced by F.M. Sokhatsky (for example [5]).Definition 3. Let k > 1, s ≥ 1, n = k · s+1 and let (Q,A) be an n−groupoid.Then, we say that (Q,A) is a P–polyagroup of the type (s, n − 1) iff it is aPs−associative n−groupoid and an n−quasigroup.A notion of a polyagroup was introduced by F.M. Sokhatsky (for example [6]).2. Auxiliary propositionsProposition 2 [10]. Let n ≥ 2 and let (Q,A) be an n−groupoid. Furthermore,let the < 1, n > −associative law hold in (Q,A), and let for every an1 ∈ Q therebe at least one x ∈ Q and at least one y ∈ Q such that the following equalitiesA(an−11 , x) = an and A(y, an−11 ) = an hold. Then, there are mappings e and−1respectively of the sets Qn−2 and Qn−1 into the set Q such that the following lawsA(e(an−21 ), an−21 , x) = x, A(x, an−21 , e(an−21 )) = x,A((an−21 , x)−1, an−21 , x) = e(an−21 ), A(x, an−21 , (an−21 , x)−1) = e(an−21 ),On NP - polyagroups 155A((an−21 , a)−1, an−21 , A(a, an−21 , x)) = x andA(A(x, an−21 , a), an−21 , (an−21 , a)−1) = xhold in the algebra (Q, {A,−1 , e}).(See, also [11].)Proposition 3. Let k > 1, s ≥ 1, n = k ·s+1 and let (Q,A) be an n−groupoid.Also, let(a) the < 1, s+ 1 > −associative [< (k − 1) · s+ 1, k · s+ 1 > −associative ] lawhold in the (Q,A); and(b) for every x, y, an−11 ∈ Q the following implication holdsA(x, an−11 ) = A(y, an−11 )⇒ x = y[ A(an−11 , x) = A(an−11 , y)⇒ x = y ].Then (Q,A) is a Ps−associative n−groupoid.Sketch of the proof.A(A(xn1 ), x2n−1n+1 ) = A(xs1, A(xs+ns+1 ), x2n−1s+n+1)⇒ A(ys1, A(A(xn1 ), x2n−1n+1 ), yn−1s+1 )= A(ys1, A(xs1, A(xs+ns+1 ), x2n−1s+n+1), yn−1s+1 )⇒ A(A(ys1, A(xn1 ), x2n−1−sn+1 ), x2n−12n−s, yn−1s+1 )= A(A(ys1, xs1, A(xs+ns+1 ), x2n−1−ss+n+1 ), x2n−12n−s, yn−1s+1 )⇒ A(ys1, A(xn1 ), x2n−1−sn+1 )= A(ys1, xs1, A(xs+ns+1 ), x2n−1−ss+n+1 ).(See, also [10,11].) ✷3. ResultsDefinition 4. Let k > 1, s ≥ 1, n = k · s + 1 and let (Q,A) be a Ps−associativen−groupoid. We shall say that (Q,A) is a near–P–polyagroup (briefly: NP–polyagroup) of the type (s, n − 1) iff for every i ∈ {t · s + 1|t ∈ {0, 1, . . . , k}}and for all an1 ∈ Q there is exactly one xi ∈ Q such that the equalityA(ai−11 , xi, an−1i ) = anholds.Remark 4. Every P–polyagroup of the type (s, n − 1) is an NP–polyagroup ofthe type (s, n− 1).Example 1. Let (Q, ·) be a group and let α be a mapping of the set Q into theset Q. Let, also, for each x51 ∈ QA(x51)def= x1 · α(x2) · x3 · α(x4) · x5.Then (Q,A) is an NP–polyagroup of the type (2,4). Moreover, if α is not a permu-tation of the set Q, then (Q,A) is not a 5−quasigroup.Theorem 1. Let k > 1, s ≥ 1, n = k · s+ 1 and let (Q,A) be an n−groupoid.Then, (Q,A) is an NP–polyagroup of the type (s, n− 1) iff there are mappings−1 and e respectively of the sets Qn−1 and Qn−2 into the set Q such that thefollowing laws hold in the algebra (Q, {A,−1 , e}) [ of the type < n, n− 1, n− 2 > ]:156 J.Usˇan and R.Galic´(i) A(A(xn1 ), x2n−1n+1 ) = A(xs1, A(xs+ns+1 ), x2n−1s+n+1),(ii) A(x, an−21 , e(an−21 )) = x and(iii) A(a, an−21 , (an−21 , a)−1) = e(an−21 ).[See, also Proposition1, Remark 2 and Theorem2]Proof. 1)⇒: Let (Q,A) be an NP–polyagroup of the type (s, n− 1). Then, byProposition 2, there is an algebra (Q, {A,−1 , e}) of the type < n, n− 1, n− 2 > inwhich the laws (i)− (iii) hold.2)⇐: Let (Q, {A,−1 , e}) be an algebra of the type < n, n− 1, n− 2 > in whichthe laws (i) − (iii) hold. We prove respectively that in that case the followingstatements hold:1◦ For every x, y, an−11 ∈ Q the following implication holdsA(x, an−11 ) = A(y, an−11 )⇒ x = y.2◦ (Q,A) is a Ps−associative n−groupoid.3◦ (∀ai ∈ Q)n−21 (∀x ∈ Q)A(e(an−21 ), an−21 , x) = x.4◦ (∀ai ∈ Q)n−21 (∀x ∈ Q)A((an−21 , a)−1, an−21 , a) = e(an−21 ).5◦ For every x, y, an−11 ∈ Q the following implication holdsA(an−11 , x) = A(an−11 , y)⇒ x = y.6◦ For every x, y, an−11 ∈ Q and for all t ∈ {1, . . . , k−1} the following implicationholdsA(at·s1 , x, an−1t·s+1) = A(at·s1 , y, an−1t·s+1)⇒ x = y.7◦ For every i ∈ {t · s+ 1|t ∈ {0, 1, . . . , k}} and for all an1 ∈ Q there is at leastone xi ∈ Q such that the following equality holds A(ai−11 , xi, an−1i ) = an.The proof of the statement of 1◦ :By n ≥ 3 (: n = k · s+ 1, k > 1, s ≥ 1), we conclude that the following series ofimplications holds:A(x, as−11 , a, an−2s ) = A(y, as−11 , a, an−2s )⇒A(A(x, as−11 , a, an−2s ), as−11 , e(an−2s , as−11 ),n−2−sa , e(as−11 ,n−2−s+1a )) =A(A(y, as−11 , a, an−2s ), as−11 , e(an−2s , as−11 ),n−2−sa , e(as−11 ,n−2−s+1a ))⇒A(x, as−11 , A(a, an−2s , as−11 , e(an−2s , as−11 )),n−2−sa , e(as−11 ,n−2−s+1a )) =A(y, as−11 , A(a, an−2s , as−11 , e(an−2s , as−11 )),n−2−sa , e(as−11 ,n−2−s+1a ))⇒A(x, as−11 , a,n−2−sa , e(as−11 ,n−2−s+1a )) =A(y, as−11 , a,n−2−sa , e(as−11 ,n−2−s+1a ))⇒ x = y.The proof of the statement of 2◦ :By (i), 1◦, n ≥ 3 and by Proposition 3, we conclude that (Q,A) is a Ps−associativen−groupoid.On NP - polyagroups 157The proof of the statement of 3◦ :A(e(an−21 ), an−21 , a) = b⇒ A(A(e(an−21 ), an−21 , a), an−21 , (an−21 , a)−1)= A(b, an−21 , (an−21 , a)−1)⇒ A(e(an−21 ), an−21 , A(a, an−21 , (an−21 , a)−1))= A(b, an−21 , (an−21 , a)−1)⇒ A(e(an−21 ), an−21 , e(an−21 ))= A(b, an−21 , (an−21 , a)−1)⇒ e(an−21 ) = A(b, an−21 , (an−21 , a)−1)⇒A(a, an−21 , (an−21 , a)−1) = A(b, an−21 , (an−21 , a)−1)⇒ a = b[: 2◦, (iii), (ii), (iii), 1◦].The proof of the proof of 4◦ :A((an−21 , a)−1, an−21 , a) = b⇒ A(A((an−21 , a)−1, an−21 , a), an−21 , (an−21 , a)−1)= A(b, an−21 , (an−21 , a)−1)⇒ A((an−21 , a)−1, an−21 , A(a, an−21 , (an−21 , a)−1))= A(b, an−21 , (an−21 , a)−1)⇒ A((an−21 , a)−1, an−21 , e(an−21 ))= A(b, an−21 , (an−21 , a)−1)⇒ A(e(an−21 ), an−21 , (an−21 , a)−1)= A(b, an−21 , (an−21 , a)−1)⇒ e(an−21 ) = b[: 2◦, (iii), (ii), 3◦, 1◦].The proof of the statement of 5◦ :Since the < 1, n > −associative law [:2◦] as well as the statements 4◦ and 3◦hold in (Q,A), we conclude that for every x, y, a ∈ Q and for every sequence an−21over Q the following series of implication holds:A(a, an−21 , x) = A(a, an−21 , y)⇒ A((an−21 , a)−1, an−21 , A(a, an−21 , x))= A((an−21 , a)−1, an−21 , A(a, an−21 , y))⇒ A(A((an−21 , a)−1, an−21 , a), an−21 , x)= A(A((an−21 , a)−1, an−21 , a), an−21 , y)⇒ A(e(an−21 ), an−21 , x)= A(e(an−21 ), an−21 , y) = x = y.The proof of the proof of 6◦ :A(at·s1 , x, ak·st·s+1) = A(at·s1 , x, ak·st·s+1)⇒ A(b(k−t)·s1 , A(at·s1 , x, ak·st·s+1), bk·s(k−t)·s+1)= A(b(k−t)·s1 , A(at·s1 , y, ak·st·s+1), bk·s(k−t)·s+1)⇒A(A(b(k−t)·s1 , at·s1 , x), ak·st·s+1, bk·s(k−t)·s+1) = A(A(b(k−t)·s1 , at·s1 , y), ak·st·s+1, bk·s(k−t)·s+1)⇒ A(b(k−t)·s1 , at·s1 , x) = A(b(k−t)·s1 , at·s1 , y)⇒ x = y[:2◦, 1◦, 5◦].The proof of the proof of 7◦ :a) t = 0 : A(x, an−21 , a) = b⇔A(A(x, an−21 , a), an−21 , (an−21 , a)−1) = A(b, an−21 , (an−21 , a)−1)⇔A(x, an−21 , A(a, an−21 , (an−21 , a)−1)) = A(b, an−21 , (an−21 , a)−1)⇔A(x, an−21 , e(an−21 )) = A(b, an−21 , (an−21 , a)−1)⇔x = A(b, an−21 , (an−21 , a)−1)[:2◦, (iii), (ii)].158 J.Usˇan and R.Galic´b) t = k : A(a, an−21 , x) = b⇔A((an−21 , a)−1, an−21 , A(a, an−21 , x)) = A((an−21 , a)−1, an−21 , b)⇔A(A(an−21 , a)−1, an−21 , a), an−21 , x)) = A((an−21 , a)−1, an−21 , b)⇔A(e(an−21 ), an−21 , x) = A((an−21 , a)−1, an−21 , b)⇔x = A((an−21 , a)−1, an−21 , b)[:2◦, 4◦, 3◦].c) 0 < t < k : A(at·s1 , x, ak·st·s+1) = b⇔ A(b(k−t)·s1 , A(at·s1 , x, ak·st·s+1), bk·s(k−t)·s+1)= A(b(k−t)·s1 , b, bk·s(k−t)·s+1)⇔A(A(b(k−t)·s1 , at·s1 , x), ak·st·s+1, bk·s(k−t)·s+1)= A(b(k−t)·s1 , b, bk·s(k−t)·s+1)[:6◦, 2◦]. ✷By a simple imitation of the proof of Theorem 1 it is possible to prove that thefollowing proposition holds:Theorem 2. Let k > 1, s ≥ 1, n = k · s+ 1 and let (Q,A) be an n−groupoid.Then, (Q,A) is an NP–polyagroup of the type (s, n− 1) iff there are mappings−1 and e respectively of the sets Qn−1 and Qn−2 into the set Q such that thefollowing laws hold in the algebra (Q, {A,−1 , e}) [ of the type < n, n− 1, n− 2 >]:(i) A(x(k−1)·s1 , A(x(k−1)·s+n(k−1)·s+1 ), x2n−1(k−1)·s+n+1) = A(xk·s1 , A(x2n−1k·s+1)),(ii) A(e(an−21 ), an−21 , x) = x and(iii) A((an−21 , a)−1, an−21 , a) = e(an−21 ).Similarly, it is possible to prove that the following proposition holds. (See, also[10,11].)Theorem 3. Let k > 1 s ≥ 1, n = k · s + 1 and let (Q,A) be an n−groupoid.Then, (Q,A) is an NP–polyagroup of the type (s, n− 1) iff there are mappings−1 and e respectively of the sets Qn−1 and Qn−2 into the set Q such that thefollowing laws hold in the algebra (Q, {A,−1 , e}) [ of the type < n, n− 1, n− 2 >]:(1) (i) from Theorem1 or (i) from Theorem2;(2) (ii) from Theorem1; and(3) A((an−21 , a)−1, an−21 , A(a, an−21 , x)) = x.Theorem 4. Let k > 1 s ≥ 1, n = k · s + 1 and let (Q,A) be an n−groupoid.Then, (Q,A) is an NP–polyagroup of the type (s, n− 1) iff there are mappings−1 and e respectively of the sets Qn−1 and Qn−2 into the set Q such that thefollowing laws hold in the algebra (Q, {A,−1 , e}) [ of the type < n, n− 1, n− 2 >]:(1) (i) from Theorem1 or (i) from Theorem2;(2) (ii) from Theorem2; and(3) A(A(x, an−21 , a), an−21 , (an−21 , a)−1) = x.References[1] W.Do¨rnte, Untersuchengen u¨ber einen verallgemeinerten Gruppenbegriff,Math. Z. 29(1928), 1-19.On NP - polyagroups 159[2] R.H.Bruck, A survey of binary systems, Springer-Verlag, Berlin-Heidelberg-New York, 1971.[3] V.D.Belousov, n−ary quasigroups, Stiinta, Kishinev, 1972 (In Russian.)[4] A.G.Kurosh, General algebra, (lectures 1969–1970), Nauka, Moscow, 1974(In Russian.)[5] F.M. Sokhatsky, On the associativity of multiplace operations, Quasigroupsand Related Systems 4(1997), 51–66.[6] F.M. Sokhatsky, O.Yurevych, Invertible elements in associates and semi-groups 2, Quasigroups and Related Systems 6(1999), 61–70.[7] J.Usˇan, Neutral operations of n-groupoids, Rev. of Research, Fac. of Sci. Univ.of Novi Sad, Math. Ser. 18(1988), 117-126 (In Russian.)[8] J.Usˇan, A comment on n-groups, Rev. of Research, Fac. of Sci. Univ. of NoviSad, Math. Ser. 24(1994), 281-288.[9] J.Usˇan, On n-groups with {i, j}-neutral operation for {i, j} 	= {1, n}, Rev. ofResearch, Fac. of Sci. Univ. of Novi Sad, Math. Ser. 25(1995), 167-178.[10] J.Usˇan, n-groups as variety of type 〈n, n− 1, n− 2〉 , in: Algebra and ModelTheory, (A.G. Pinus and K.N.Ponomaryov, eds.), Novosibirsk, 1997, 182-208.[11] J.Usˇan, Note on (n,m)-groups, Math. Moravica 3(1999), 127–139.

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