In this paper a class of (m,n)-rings with a left and right zero is described as a variety of algebras of type 

J. Ušan

R. Galić

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Mathematical Communications 5(2000), 133-141 133(m,n)-rings as algebras with only one operationJanez Usˇan∗ and RadoslavGalic´†Abstract. In this paper a class of (m,n)-rings with a left and rightzero is described as a variety of algebras of type < 3m+ n− 5, 0 > .Key words: n−groupoid, n−semigroup, n−quasigroup, n−group,{i, j}−neutral operation, inversing operation,(m,n)−ringAMS subject classifications: 20N15Received June 10, 2000 Accepted October 9, 20001. PreliminariesA notion of an n−group was introduced by W. Do¨rnte in [7] as a generalization ofthe notion of a group. See, also [3], [1], [10].Definition 1. Let n ≥ 2 and let (Q,A) be an n−groupoid. We say that (Q,A) is aDo¨rnte n−group [briefly: n−group] iff it is an n−semigroup and an n−quasigroupas well.Proposition 1. (see [15]) 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 −1 and 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(iii) there are mappings −1 and 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 >]∗Institute of Mathematics, University of Novi Sad, Trg D.Obradovic´a 4, 21 000 Novi Sad,Yugoslavia†Faculty of Electrical Engineering, University of Osijek, Kneza Trpimira 2B, HR-31 000 Osijek,Croatia, e-mail: galic@etfos.hr134 J.Usˇan and R.Galic´(a) A(A(xn1 ), x2n−1n+1 ) = A(x1, A(xn+12 ), x2n−1n+2 ),(b) A(x, an−21 , e(an−21 )) = x and(c) A(a, an−21 , (an−21 , a)−1) = e(an−21 ).Remark 1. e is a {1, n}−neutral operation of an n−grupoid (Q,A) iff algebra(Q, {A, e}) of type < n, n − 2 > satisfies the laws (b) and (b) from Proposition1(cf. [12]). The notion of {i, j}−neutral operation (i, j ∈ {1, . . . , n}, i < j) ofan n−groupoid is defined in a similar way (cf. [12]). Every n−groupoid has atmost one {i, j}−neutral operation (cf. [12]). In every n−group, n ≥ 2, there isa {1, n}−neutral operation (cf. [12]). There are n−groups without {i, j}−neutraloperations with {i, j} = {1, n} (cf. [14]). In [14], n−groups with {i, j}−neutraloperations, for {i, j} = {1, n} are described. Operation −1 from Proposition1 isa generalization of the inversing operation in a group. In fact, if (Q,A) is ann−group, n ≥ 2, then for every a ∈ Q and for every 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 ) (cf. [13]). (For n = 2, a−1 = E(a); a−1 is theinverse element of the element a with respect to the neutral element e(∅) of thegroup (Q,A).)Proposition 2. (see [14]) Let n ≥ 3, let (Q,A) be an n-group and e its {1, n}-neutral operation. Then the following statements are equivalent:(i) (Q,A) is a commutative n-group,(ii) e is an {i, j}-neutral operation of the n-group (Q,A) for every {i, j}⊆{1, . . . , n},i < j.Proposition 3. (see [16]) Let (Q,A) be an m-group, −1 its inversing operation,e its {1,m}-neutral operation and let m ≥ 2. Also let(o) −1A(x, am−21 , y) = zdef⇔A(z, am−21 , y) = xfor x, y, z ∈ Q and for every sequence am−21 over Q. Then, for all x, y, z ∈ Q andfor every sequence am−21 over Q the following equalities hold(1) −1A(x, am−21 , y) = A(x, am−21 , (am−21 , y)−1),(2) e(am−21 ) =−1A(z, am−21 , z),(3) (am−21 , x)−1 =−1A(−1A(z, am−21 , z), am−21 , x) and(4) A(x, am−21 , y) =−1A(x, am−21 ,−1A(−1A(z, am−21 , z), am−21 , y)).Proposition 4. (see [16]) Let n ≥ 2 and let (Q,B) be an n−groupoid. Let alsothe following lawsB(B(x, z, bn−21 ), B(y, an−21 , z), an−21 ) = B(x, y, bn−21 ) andB(a, cn−21 , B(B(B(z, cn−21 , z), cn−21 , b), cn−21 , B(B(z, cn−21 , z), cn−21 , a))) = bhold in the n−groupoid (Q,B). Then, there is an n−group (Q,A) such that thefollowing equality holds −1A = B.(m,n)-rings as algebras with only one operation 135Definition 2. ( see [2],[5],[8] and [9]) Let (Q,A) be a commutative m−groupand let m ≥ 2. Let also (Q,M) be an n−groupoid (n−semigroup in [2], [4]) andlet n ≥ 2. We say that (Q,A,M) is an (m,n)−ring iff for every i ∈ {1, . . . , n} andfor every an−11 , bm1 ∈ Q the following equality holdsM(ai−11 , A(bm1 ), an−1i ) = A(M(ai−11 , bj, an−1i )∣∣mj=1).Proposition 5. Let (Q,A,M) be an (m,n)−ring. Then, there is at most oneelement o ∈ Q such that for every an−1 ∈ Q the following equalities holdM(o, an−11 ) = o and M(an−11 , o) = o.See [4].Proposition 6. Let (Q,A,M) be an (m,n)−ring and O the {1,m}−neutral oper-ation of the m−group (Q,A). Also, let o be the element of the set Q such that(oˆ) M(o, an−11 ) =M(an−11 , o) = ofor all an−11 ∈ Q. Then the following equality holdsO(m−2o ) = o.See [4] [O(m−2o ) = o for m>2 ; a is skewed to a [7]].Sketch of the proof.a) M(A(mo ), an−11 ) = A(mM(o, an−11 |) = A(mo ),M(an−11 , A(mo )) = A(mM(an−11 , o|) = A(mo ) [ : Definition 2., (oˆ) ].b) By a) and Proposition 5., we conclude that the following equality holds A(mo ) = o.c) A(O(m−2o ),m−2o , o) = o , A(o,m−2o , o) = o [ : Proposition 1., Remark 1.,b) ].d) By c), Definition 1., Proposition 1. and Remark 1., we conclude that the followingequality holds O(m−2o ) = o. (For m = 2 :m−2a = ∅.)Remark 2.(a) Note that element o determined in Proposition5. (Proposition6.) is called aleft and right zero, respectively, or a two-side zero.(b) An element z ∈ Q is called zero of (Q,A,M) iff for each i ∈ {1, . . . , n} andfor every an−11 ∈ Q the following equality holds M(ai−11 , z, an−1i ) = z.(c) Let (Q,A,M) be an (m,n)-ring, o its two-side zero and m>2. Then, by Propo-sition 1., Proposition6. and Definition 2., we conclude that for each x ∈ Qand for each i ∈ {1, . . . ,m} the following equality holds A(i−1o , x,m−io ) = x.Whence, we conclude that there is a group (Q,+) such that for every xm1 ∈ Qthe following equality holds A(xm1 ) = x1 + ...+ xm (cf. [6]).136 J.Usˇan and R.Galic´2. ResultsTheorem 1. Let (Q, T, o) be an algebra of the type < 3m + n − 5, 0 >, and letm,n ≥ 2. Also let(a) α(x, am−21 , y)def= T(x, T (T (2m−1o , am−21 ,n−2o ), y,2m−3o , am−21 ,n−2o ),2m−3o , am−21 ,n−2o),(b) β(x, am−21 , y)def= T(x, y,2m−3o , am−21 ,n−2o)and(c) γ(x, bn−21 , y)def= T(T (x, o, y,3m−6o , bn−21 ), x,3m+n−7o)for every x, y, am−21 , bn−21 ∈ Q.Furthermore, let the following laws(i) β(β(x, z, bm−21 ),β(y, am−21 , z), am−21 ) = β(x, y, bm−21 ),(ii) β(a, cm−21 ,β(β(β(z, cm−21 , z), cm−21 , b), cm−21 ,β(β(z, cm−21 , z), cm−21 , a)))= b,(iii) α(xϕ(1), . . . , xϕ(m)) = α(xm1 ) for all permutations ϕ on {1, . . . ,m},(iv) γ(xi−11 ,β(ym1 ), xn−1i ) = β(γ(xi−11 , yj , xn−1i )|mj=1) for all i ∈ {1, . . . , n},(v) T (x,3m−4o , bn−21 ) = x,(vi) T (o, o, y,3m−6o , bn−21 ) = o and(vii) T (x, x,3m+n−7o ) = ohold in the algebra (Q, T, o) of the type < 3m + n − 5, 0 > . Then (Q,α,γ) is an(m,n)-ring and o is its two-side zero.Proof.1) By (b), (i), (ii) and Proposition 4., we conclude that there is an m−group(Q,A) such that the following equality holds −1A = β.2) A = α. The sketch of the proof:α(x, am−21 , y)= T (x, T (T (2m−1o , am−21 ,n−2o ), y,2m−3o , am−21 ,n−2o ),2m−3o , am−21 ,n−2o )= T (x, T (β(o, am−21 , o), y,2m−3o , am−21 ,n−2o ),2m−3o , am−21 ,n−2o )= T (x,β(β(o, am−21 , o), am−21 , y),2m−3o , am−21 ,n−2o )= β(x, am−21 ,β(β(o, am−21 , o), am−21 , y))= −1A(x, am−21 ,−1A(−1A(o, am−21 , o), am−21 , y)) = A(x, am−21 , y)[ : (a), (b), 1), Proposition 3. ].3) By 1), 2), (iii), (iv) andDefinition 2, we conclude that (Q,α,γ) is an (m,n)−ring.(m,n)-rings as algebras with only one operation 1374) For all an−11 ∈ Q the following equalities holdγ(o, an−11 ) = o and γ(an−11 , o) = o.The sketch of the proof:γ(o, bn−21 , y) = T (T (o, o, y,3m−6o , bn−21 ), o,3m+n−7o )= T (o, o,3m+n−7o ) = o;γ(x, bn−21 , o) = T (T (x, o, o,3m−6o , bn−21 ), x,3m+n−7o )= T (x, x,3m+n−7o ) = o;[ : (c),(v),(vi),(vii) ]. ✷Theorem 2. Let (Q,A,M) be an (m,n)−ring, O be the {1,m}−neutral operationof the m−group (Q,A), and – the inversing operation of the m−group (Q,A). Also,let o be the element of the set Q such that(a) M(o, an−11 ) =M(an−11 , o) = o for all an−11 ∈ Q (i.e., let o be a two-side zero).(o) T (x, y, z, am−21 , bm−21 , cm−21 , dn−21 )def=A(−1A(M(x, dn−21 , z), am−21 ,M(y, dn−21 , z)), bm−21 ,−1A(x, cm−21 , y)).1for each x, y, z, am−21 , bm−21 , cm−21 , dn−21 ∈ QThen, the following identities hold:(1) O(bm−21 ) = T (x, x, y,2am−21∣∣, bm−21 , cn−21 );(2) – (bm−21 , x) = T (T (2u, v,2am−21∣∣, bm−21 , cn−21 ), x, o,2dm−21∣∣, bm−21 , en−21 );(3) −1A(x, bm−21 , y) = T (x, y, o,2am−21∣∣, bm−21 , cn−21 );(4) A(x, bm−21 , , y) =T (x, T (T (2u, v,2am−21∣∣, bm−21 , cn−21 ), y, o,2dm−21∣∣, bm−21 , en−21 ), o,2pm−21∣∣, bm−21 , qn−21 );(5) M(x, bn−21 , y) = T (T (x, o, y,m−2o , am−21 ,m−2o , bn−21 ), x, o,2cm−21∣∣, am−21 , dn−21 );(6) T (x,2o,2am−21∣∣,m−2o , bn−21 ) = x;(7) T (2o, x,2am−21∣∣,m−2o , bn−21 ) = o; and(8) T (2x, y,2am−21∣∣,m−2o , bn−21 ) = o.1cf. [11]/(7) or [10]/10/(11)138 J.Usˇan and R.Galic´Sketch of the proof.1) T (x, x, y,2am−21∣∣, bm−21 , cn−21 )= A(−1A(M(x, cn−21 , y), am−21 ,M(x, cn−21 , y)), am−21 ,−1A(x, am−21 , x))= A(O(am−21 ), am−21 ,−1A(x, am−21 , x)) =−1A(x, am−21 , x)) = O(am−21 )[ : (o), Proposition 3., Proposition 1., Remark 1. ].2) T (T (2u, v,2am−21∣∣, bm−21 , cn−21 ), x, o,2dm−21∣∣, bm−21 , en−21 )= T (O(bm−21 ), x, o,2dm−21∣∣, bm−21 , en−21 )= A(−1A(M(O(bm−21 ), en−21 , o), dm−21 , M(x, en−21 , o)), dm−21 , A(O(bm−21 ), bm−21 , x))= A(−1A(o, dm−21 , , o)dm−21 , , A(O(bm−21 ), bm−21 , − (bm−21 , x)))= A(O(dm−21 ), dm−21 , − (bm−21 , x)) = −(bm−21 , x)[ : (o), 1), (a), Proposition 3., Proposition 1., Remark 1. ].3) T (x, y, o,2am−21∣∣, bm−21 , cn−21 )= A(−1A(M(x, cn−21 , o), am−21 ,M(y, cn−21 , o)), am−21 ,−1A(x, bm−21 , y))= A(−1A(o, am−21 , o), am−21 ,−1A(x, bm−21 , y))= A(O(am−21 ), am−21 ,−1A(x, bm−21 , y))=−1A(x, bm−21 , y))[ : (o), (a), Proposition 3., Proposition 1., Remark 1. ].4) T (x, T (T (2u, v,2am−21∣∣, bm−21 , cn−21 ), y, o,2dm−21∣∣, bm−21 , en−21 ), o,2pm−21∣∣, bm−21 , qn−21 )= T (x,−(bm−21 , y), o,2pm−21∣∣, bm−21 , qn−21 )= A(−1A(M(x, qn−21 , o), pm−21 , M(−(bm−21 , y), qn−21 , o)), pm−21 ,−1A(x, bm−21 , −(bm−21 , y)))= A(−1A(o, pm−21 , o), pm−21 ,−1A(x, bm−21 ,−(bm−21 , y)))= A(O(pm−21 ), pm−21 ,−1A(x, bm−21 ,−(bm−21 , y)))= −1A(x, bm−21 ,−(bm−21 , y)) = A(x, bm−21 ,−(bm−21 ,−(bm−21 , y)))= A(x, bm−21 , y)[ : (o), 2), (a), 3., Definition 1., Remark 1. ].51) M(x, bn−21 , y) =−1A(M(x, bn−21 , y),m−2o ,O(m−2o ))= −1A(M(x, bn−21 , y),m−2o , o) =−1A(M(x, bn−21 , y),m−2o ,M(o, bn−21 , y))= A(−1A(M(x, bn−21 , y),m−2o ,M(o, bn−21 , y)), am−21 ,O(am−21 ))= A(−1A(M(x, bn−21 , y),m−2o ,M(o, bn−21 , y)), am−21 , A(x, am−21 ,−(am−21 , x)))(m,n)-rings as algebras with only one operation 139= A(A(−1A(M(x, bn−21 , y),m−2o ,M(o, bn−21 , y)), am−21 , x), am−21 ,−(am−21 , x))= A(A(−1A(M(x, bn−21 , y),m−2o ,M(o, bn−21 , y)), am−21 ,−1A(x,m−2o , o)), am−21 , − (am−21 , x))= −1A(A(−1A(M(x, bn−21 , y),m−2o ,M(o, bn−21 , y)), am−21 ,−1A(x,m−2o , o)), am−21 , x)= −1A(T (x, o, y,m−2o , am−21 ,m−2o , bn−21 ), am−21 , x)[ : Proposition 1., Remark 1., Proposition 6., (a), (o) ].52) Xdef= T (x, o, y,m−2o , am−21 ,m−2o , bn−21 ).53) M(x, bn−21 , y) =−1A(X, am−21 , x) = A(O(cm−21 ), cm−21 ,−1A(X, am−21 , x))= A(−1A(o, cm−21 , o), cm−21 ,−1A(X, am−21 , x))= A(−1A(M(X, dn−21 , o), cm−21 ,M(x, dn−21 , o)), cm−21 ,−1A(X, am−21 , x))= T (X,x, o,2cm−21∣∣, am−21 , dn−21 )= T (T (x, o, y,m−2o , am−21 ,m−2o , bn−21 ), x, o,2cm−21∣∣, am−21 , dn−21 )[ : 51), 52), Proposition 1., Remark 1., Proposition 3., (a), (o) ].6) T (x,2o,2am−21∣∣,m−2o , bn−21 )= A(−1A(M(x, bn−21 , o), am−21 ,M(o, bn−21 , o)), am−21 ,−1A(x,m−2o , o))= A(−1A(o, am−21 , o), am−21 ,−1A(x,m−2o , o))= A(O(am−21 ), am−21 ,−1A(x,m−2o , o)) =−1A(x,m−2o , o)= −1A(x,m−2o ,O(m−2o )) = x;−1A(x,m−2o ,O(m−2o )) = z ⇔A(z,m−2o ,O(m−2o )) = x⇔ z = x[ : (o), (a), Proposition 3., Proposition 1., Remark 1., Proposition 6. ].7) T (2o, x,2am−21∣∣,m−2o , bn−21 )= A(−1A(M(o, bn−21 , x), am−21 ,M(o, bn−21 , x)), am−21 ,−1A(o,m−2o , o))= A(O(am−21 ), am−21 ,−1A(o,m−2o , o)) =−1A(o,m−2o , o) = O(m−2o ) = o[ : (o), Proposition 3., Proposition 1., Remark 1., Proposition 6. ].8) T (2x, y,2am−21∣∣,m−2o , bn−21 )= A(−1A(M(x, bn−21 , y), am−21 ,M(x, bn−21 , y)), am−21 ,−1A(x,m−2o , x))= A(O(am−21 ), am−21 ,−1A(x,m−2o , x)) =−1A(x,m−2o , x) = O(m−2o ) = o[ : (o), Proposition 3., Proposition 1., Remark 1., Proposition 6. ]. ✷140 J.Usˇan and R.Galic´Remark 3. The operation O has been described in Theorem2. by using only theoperation T. Bearing in mind Proposition6., (1) from Theorem2., as well as theconvention that a◦1 = ∅, we find that for each x, y, cn−21 ∈ Q, (1) reduces to thefollowing equality(m) o = T (x, x, y, cn−21 ).Hence, bearing in mind Theorem2., we find out that in (2, n)−rings the operationsA,−1A,− and M can also be described by using just one (n+1)− ary operation. Inaddition, in the case m = 2, the constant o ∈ Q can be eliminated from equalities(6)–(8) in Theorem2. by using (m).In [11] rings [(2,2)–rings] have been described as 3–groupoids with one law.References[1] V.D.Belousov, n−ary Quasigroups, Stiinca, Kishinev 1972 (in Russian).[2] D.Boccioni, Caratterizzazione di una classe di anelli generalizzati, Rend.Sem. Mat. Univ. Padova 35(1965), 116–127.[3] R.H.Bruck, A Survey of Binary Systems, Springer - Verlag, Berlin - Heidel-berg - New York, 1971.[4] G.Crombez, On (n,m)−rings, Abh. Math. Sem. Univ. 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(m,n)-rings as algebras with only one operation

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