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				 \begin{document}$$[a , b]=a^{-1}b^{-1}ab$$\end{document}[a , b]=a-1b-1ab of two elements \documentclass[12pt]{minimal } 
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				 \begin{document}$$a$$\end{document}a and \documentclass[12pt]{minimal } 
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				 \begin{document}$$b$$\end{document}b of a group \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf g $ $ \end{document}g can be seen as a  measure  how far are \documentclass[12pt]{minimal } 
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				 \begin{document}$$a$$\end{document}a and \documentclass[12pt]{minimal } 
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				 \begin{document}$$b$$\end{document}b from commuting according to the group operation of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf g $ $ \end{document}g .
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				 \begin{document}$$[g , g]$$\end{document}[g , g ] generated by all such commutators  measures  how far is the group \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf g $ $ \end{document}g from an abelian group .
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				 \begin{document}$$[g , g]=0$$\end{document}[g , g]=0 if and only if \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf g $ $ \end{document}g is abelian .
the concept of commutators of normal subgroups has been generalized to a binary operation of the congruence lattice of arbitrary algebras in congruence modular varieties by smith , freese , mckenzie , hagemann , herrmann, ... one can find more details in .
a malcev term of an algebra \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a is a ternary term operation \documentclass[12pt]{minimal } 
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				 \begin{document}$$d$$\end{document}d of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a that satisfies \documentclass[12pt]{minimal } 
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				 \begin{document}$$d(x , y , y)=d(y , y , x)=x$$\end{document}d(x , y , y)=d(y , y , x)=x for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$x , y\in a$$\end{document}x , ya . an algebra with a malcev term we call a malcev algebra .
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				 \begin{document}$$=0$$\end{document}=0 in malcev algebras means has been answered by gumm , hagemann and herrmann , see [ 4 , theorem 13.4 ] .
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				 \begin{document}$$(=0)$$\end{document}(=0 ) if and only if it is polynomially equivalent to a module over a ring . here , we call two algebras \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf b $ $ \end{document}b on the same domain polynomially equivalent if they have the same set of polynomial operations . in 2001 , a. bulatov has generalized the binary commutator operations to \documentclass[12pt]{minimal } 
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				 \begin{document}$$n$$\end{document}n - ary commutator operations \documentclass[12pt]{minimal } 
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				 \begin{document}$$[\bullet , \bullet , \ldots , \bullet ] $ $ \end{document}[,, 
, ] on congruence lattices of malcev algebras , for each \documentclass[12pt]{minimal } 
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				 \begin{document}$$n\in \mathbb n $ $ \end{document}nn , see .
the aim of this note is to answer what \documentclass[12pt]{minimal } 
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				 \begin{document}$$=0$$\end{document}=0 means in malcev algebras .
we want to characterize all malcev algebras with such a property , because it could be expected that \documentclass[12pt]{minimal } 
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				 \begin{document}$$=0$$\end{document}=0 implies that the algebra is polynomially equivalent to a well studied structure .
according to the property ( hc3 ) in   we know that \documentclass[12pt]{minimal } 
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				 \begin{document}$$\le $ $ \end{document}. therefore , we have that each abelian malcev algebra satisfies the condition \documentclass[12pt]{minimal } 
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				 \begin{document}$$=0$$\end{document}=0 . in theorem  3.3
we prove that every malcev algebra with \documentclass[12pt]{minimal } 
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				 \begin{document}$$=0$$\end{document}=0 is an expanded group . in accordance with [ 2 , definition 7.1 ] , malcev algebras that satisfy \documentclass[12pt]{minimal } 
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				 \begin{document}$$=0$$\end{document}=0 are called 2-supernilpotent . as it has been defined in   algebras that satisfy \documentclass[12pt]{minimal } 
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				 \begin{document}$$[1,]=0$$\end{document}[1,]=0 are called 2-nilpotent . in expanded groups we deal with ideals rather then with congruences , because the corresponding commutator operations act in the same way according to [ 2 , corollary 6.12 ] . an expanded group \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v is 2-nilpotent if \documentclass[12pt]{minimal } 
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				 \begin{document}$$[v,[v , v]]=0$$\end{document}[v,[v , v]]=0 and 2-supernilpotent if \documentclass[12pt]{minimal } 
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				 \begin{document}$$[v , v , v]=0$$\end{document}[v , v , v]=0 .
a polynomial of an algebra \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a is an operation obtained by composition of projections , fundamental and constant operations of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a , see [ 6 , definition 4.4 ] .
the set of all \documentclass[12pt]{minimal } 
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				 \begin{document}$$n$$\end{document}n - ary polynomials of an algebra \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a we denote by \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathsf{pol}_n\mathbf a $ $ \end{document}polna for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$n\in \mathbb n $ $ \end{document}nn .   let \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v = ( v,+,-,0,f)$$\end{document}v=(v,+,-,0,f ) be an expanded group and let \documentclass[12pt]{minimal } 
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				 \begin{document}$$n\in \mathbb n $ $ \end{document}nn .
we call an \documentclass[12pt]{minimal } 
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				 \begin{document}$$n$$\end{document}n - ary polynomial \documentclass[12pt]{minimal } 
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				 \begin{document}$$f$$\end{document}f of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}vabsorbing if \documentclass[12pt]{minimal } 
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				 \begin{document}$$f(a_{1},\ldots , a_{n})=0$$\end{document}f(a1, 
,an)=0 whenever there exists an \documentclass[12pt]{minimal } 
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				 \begin{document}$$i\in \{1,\ldots , n\}$$\end{document}i{1, 
,n } such that \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_i=0$$\end{document}ai=0 .
the set of unary absorbing polynomials of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v we denote by \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_0(\mathbf v ) $ $ \end{document}p0(v ) and the set of binary absorbing polynomials of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v we denote by \documentclass[12pt]{minimal } 
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				 \begin{document}$$cp(\mathbf v ) $ $ \end{document}cp(v ) .
more generally , for an algebra \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a and an \documentclass[12pt]{minimal } 
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				 \begin{document}$$n\in \mathbb n $ $ \end{document}nn , \documentclass[12pt]{minimal } 
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				 \begin{document}$$(a_{1},\ldots , a_{n})\in a^n$$\end{document}(a1,  ,an)an , \documentclass[12pt]{minimal } 
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				 \begin{document}$$a\in a$$\end{document}aa we say that an \documentclass[12pt]{minimal } 
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				 \begin{document}$$n$$\end{document}n - ary polynomial \documentclass[12pt]{minimal } 
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				 \begin{document}$$p$$\end{document}p is absorbing at\documentclass[12pt]{minimal } 
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				 \begin{document}$$(a_{1},\ldots , a_{n})$$\end{document}(a1,
 ,an)with value\documentclass[12pt]{minimal } 
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				 \begin{document}$$a$$\end{document}a if \documentclass[12pt]{minimal } 
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				 \begin{document}$$p(x_{1},\ldots , x_{n})=a$$\end{document}p(x1, 
,xn)=a whenever there exists an \documentclass[12pt]{minimal } 
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				 \begin{document}$$i\in \{1,\ldots , n\}$$\end{document}i{1, 
,n } such that \documentclass[12pt]{minimal } 
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				 \begin{document}$$x_i = a_i$$\end{document}xi = ai .    let \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v be an expanded group with a group reduct \documentclass[12pt]{minimal } 
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				 \begin{document}$$(v,+,-,0)$$\end{document}(v,+,-,0 ) , and let \documentclass[12pt]{minimal } 
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				 \begin{document}$$n\in \mathbb n $ $ \end{document}nn . for \documentclass[12pt]{minimal } 
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				 \begin{document}$$i\le n$$\end{document}in let \documentclass[12pt]{minimal } 
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				 \begin{document}$$\pi _ i^n : v^n\rightarrow v$$\end{document}in : vnv be the \documentclass[12pt]{minimal } 
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				 \begin{document}$$i$$\end{document}i - th projection
. then the group \documentclass[12pt]{minimal } 
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				 \begin{document}$$(\mathsf{pol}_n\mathbf{v } , + , -,0)$$\end{document}(polnv,+,-,0 ) is generated by constants and\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned}&\bigcup _ { k=1}^n\{f(\pi _ { i_1}^n,\ldots , \pi _ { i_k}^n):f \text { is a nonzero absorbing polynomial of } \mathbf v \text { of degree } k,\\&\qquad i_1,\ldots , i_k\in \{1,\ldots , n\ } \}.
\end{aligned}$$\end{document}k=1n{f(i1n,  ,ikn):fis a nonzero absorbing polynomial ofvof degreek , i1,  ,ik{1,  ,n}}.   see [ 1 , lemma 3.1 ] . in [ 1 , p. 260
] each \documentclass[12pt]{minimal } 
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				 \begin{document}$$n$$\end{document}n - ary commutator polynomial is exactly a polynomial \documentclass[12pt]{minimal } 
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				 \begin{document}$$f(\pi _ { i_1}^n,\ldots , \pi _ { i_k}^n)$$\end{document}f(i1n,  ,ikn ) where \documentclass[12pt]{minimal } 
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				 \begin{document}$$k\le n$$\end{document}kn and \documentclass[12pt]{minimal } 
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				 \begin{document}$$f$$\end{document}f is a nonzero \documentclass[12pt]{minimal } 
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				 \begin{document}$$k$$\end{document}k - ary absorbing polynomial defined in definition  2.1 or constant .
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				 \begin{document}$$\square $ $ \end{document}   let \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v = ( v,+,-,0,f)$$\end{document}v=(v,+,-,0,f ) be an expanded group such that\documentclass[12pt]{minimal } 
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				 \begin{document}$$f$$\end{document}f is the set of at most binary absorbing operations on \documentclass[12pt]{minimal } 
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				 \begin{document}$$v$$\end{document}v , every absorbing operation in \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathsf{pol}_2(\mathbf v ) $ $ \end{document}pol2(v ) is distributive with respect to \documentclass[12pt]{minimal } 
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				 \begin{document}$$+$$\end{document}+ , and\documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v is \documentclass[12pt]{minimal } 
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				 \begin{document}$$2$$\end{document}2-nilpotent.then \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v is \documentclass[12pt]{minimal } 
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				 \begin{document}$$2$$\end{document}2-supernilpotent .  
\documentclass[12pt]{minimal } 
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				 \begin{document}$$f$$\end{document}f is the set of at most binary absorbing operations on \documentclass[12pt]{minimal } 
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				 \begin{document}$$v$$\end{document}v , every absorbing operation in \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathsf{pol}_2(\mathbf v ) $ $ \end{document}pol2(v ) is distributive with respect to \documentclass[12pt]{minimal } 
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				 \begin{document}$$+$$\end{document}+ , and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v is \documentclass[12pt]{minimal } 
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				 \begin{document}$$2$$\end{document}2-nilpotent .
first , we show that for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$k\in \mathbb
n $ $ \end{document}kn and \documentclass[12pt]{minimal } 
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				 \begin{document}$$f\in \mathsf{pol}_k(\mathbf v ) $ $ \end{document}fpolk(v ) there exist \documentclass[12pt]{minimal } 
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				 \begin{document}$$c\in v$$\end{document}cv , \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_i\in p_0(\mathbf v ) $ $ \end{document}pip0(v ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$$q_{ij}\in cp(\mathbf v ) $ $ \end{document}qijcp(v ) for \documentclass[12pt]{minimal } 
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				 \begin{document}$$1\le i < j\le k$$\end{document}1i < jk such that2.1\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } f(x_{1},\ldots , x_{k})=c+p_1(x_1)+\cdots + p_k(x_k)+\sum _ { 1\le i
 ,xk)=c+p1(x1)++pk(xk)+1i < jkqij(xi , xj).let \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_k$$\end{document}pk be the set of operations as in ( 2.1 ) .
obviously \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_k\subseteq \mathsf{pol}_k(\mathbf
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				 \begin{document}$$\mathsf{pol}_k(\mathbf v ) \subseteq p_k$$\end{document}polk(v)pk , we need to show that the set \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_k$$\end{document}pk contains projections and constants and that is closed under the basic operations of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v .
projections are in \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_0(\mathbf v ) $ $ \end{document}p0(v ) and therefore in \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_k$$\end{document}pk .
constants are in \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_k$$\end{document}pk because \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_0(\mathbf v ) $ $ \end{document}p0(v ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$$cp(\mathbf v ) $ $ \end{document}cp(v ) contain zero polynomials .
clearly , \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_k$$\end{document}pk is closed under \documentclass[12pt]{minimal } 
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				 \begin{document}$$+$$\end{document}+ and \documentclass[12pt]{minimal } 
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				 \begin{document}$$-$$\end{document}- since \documentclass[12pt]{minimal } 
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				 \begin{document}$$[c , p(x_i)]=r(x_i)$$\end{document}[c , p(xi)]=r(xi ) , for some \documentclass[12pt]{minimal } 
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				 \begin{document}$$r\in p_0(\mathbf v ) $ $ \end{document}rp0(v ) , \documentclass[12pt]{minimal } 
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				 \begin{document}$$[c , q_{ij}(x_i , x_j)]=s(x_i , x_j)$$\end{document}[c , qij(xi , xj)]=s(xi , xj ) for some \documentclass[12pt]{minimal } 
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				 \begin{document}$$s\in cp(\mathbf{v } ) $ $ \end{document}scp(v ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$$[p(x_i),p(x_j)]=t(x_i , x_j)$$\end{document}[p(xi),p(xj)]=t(xi , xj ) for some \documentclass[12pt]{minimal } 
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				 \begin{document}$$t\in cp(\mathbf{v } ) $ $ \end{document}tcp(v ) .
furthermore , \documentclass[12pt]{minimal } 
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				 \begin{document}$$[q_{ij}(x_i , x_j),p_l(x_l)]$$\end{document}[qij(xi , xj),pl(xl ) ] and \documentclass[12pt]{minimal } 
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				 \begin{document}$$[q_{ij}(x_i , x_j),q_{lt}(x_l , x_t)]$$\end{document}[qij(xi , xj),qlt(xl , xt ) ] are zero polynomials because \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v is \documentclass[12pt]{minimal } 
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				 \begin{document}$$2$$\end{document}2-nilpotent .
therefore , in the sum of two operations in the form of ( 2.1 ) one can permute all summands to obtain again the form as in ( 2.1 ) , because \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_0(\mathbf v ) $ $ \end{document}p0(v ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$$cp(\mathbf v ) $ $ \end{document}cp(v ) are closed under \documentclass[12pt]{minimal } 
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				 \begin{document}$$+$$\end{document}+ and \documentclass[12pt]{minimal } 
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				 \begin{document}$$-$$\end{document}- . to show that \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_k$$\end{document}pk is closed under all \documentclass[12pt]{minimal } 
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				 \begin{document}$$q\in cp(\mathbf{v } ) $ $ \end{document}qcp(v ) we note that for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$q\in cp(\mathbf{v } ) $ $ \end{document}qcp(v ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$$f , f'\in p_k$$\end{document}f , fpk we have \documentclass[12pt]{minimal } 
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				 \begin{document}$$q(f , f')\in p_k$$\end{document}q(f , f)pk by the distributivity of \documentclass[12pt]{minimal } 
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				 \begin{document}$$q$$\end{document}q and the \documentclass[12pt]{minimal } 
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				 \begin{document}$$2$$\end{document}2-nilpotence of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v .
it remains to show that \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_k$$\end{document}pk is closed under any unary operation \documentclass[12pt]{minimal } 
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				 \begin{document}$$g\in f$$\end{document}gf .
note that \documentclass[12pt]{minimal } 
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				 \begin{document}$$g'(x , y):=g(x+y)-g(x)-g(y)$$\end{document}g(x , y):=g(x+y)-g(x)-g(y ) is in \documentclass[12pt]{minimal } 
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				 \begin{document}$$cp(\mathbf v ) $ $ \end{document}cp(v ) , hence distributive . by induction
it follows that for every \documentclass[12pt]{minimal } 
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				 \begin{document}$$n$$\end{document}n there exist \documentclass[12pt]{minimal } 
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				 \begin{document}$$c_{ij}\in cp(\mathbf v ) $ $ \end{document}cijcp(v ) such that2.2\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } g(x_1+\cdots + x_n)=g(x_1)+\cdots + g(x_n)+\sum _ { 1\le i <
j\le n}c_{ij}(x_i , x_j ) . \end{aligned}$$\end{document}g(x1++xn)=g(x1)++g(xn)+1i < jncij(xi , xj).for \documentclass[12pt]{minimal } 
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				 \begin{document}$$f$$\end{document}f as in ( 2.1 ) this yields2.3\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } gf(x_1,\ldots , x_k)&= g(c)+gp_1(x_1)+\cdots + gp_k(x_k)+\sum _ { 1\le i < j\le k}gq_{ij}(x_i , x_j)\nonumber \\&+\sum _ { 1\le i\le k}c_i(c , p_i(x_i))+\sum _ { 1\le i < j\le k}c_{ij}(p(x_i),p(x_j ) ) \end{aligned}$$\end{document}gf(x1,  ,xk)=g(c)+gp1(x1)++gpk(xk)+1i < jkgqij(xi , xj)+1ikci(c , pi(xi))+1i < jkcij(p(xi),p(xj))for some \documentclass[12pt]{minimal } 
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				 \begin{document}$$c_i , c_{ij}\in
cp(\mathbf v ) $ $ \end{document}ci , cijcp(v ) . note that the other terms vanish by \documentclass[12pt]{minimal } 
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				 \begin{document}$$[v,[v , v]]=0$$\end{document}[v,[v , v]]=0 . thus \documentclass[12pt]{minimal } 
				
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				 \begin{document}$$gf\in p_k$$\end{document}gfpk .
we have proved \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_k=\mathsf{pol}_k(\mathbf v ) $ $ \end{document}pk = polk(v ) .
now we know that for every ternary polynomial operation \documentclass[12pt]{minimal } 
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				 \begin{document}$$p$$\end{document}p on \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v there exist a \documentclass[12pt]{minimal } 
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				 \begin{document}$$c\in v$$\end{document}cv , \documentclass[12pt]{minimal } 
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				 \begin{document}$$f , g , h\in p_0(\mathbf v ) $ $ \end{document}f , g , hp0(v ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$$r , s , t\in cp(\mathbf{v } ) $ $ \end{document}r , s , tcp(v ) such that2.4\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } p(x , y , z)=c+f(x)+g(y)+h(z)+r(x , y)+s(x , z)+t(y , z ) \end{aligned}$$\end{document}p(x , y , z)=c+f(x)+g(y)+h(z)+r(x , y)+s(x , z)+t(y , z)for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$x , y , z\in v$$\end{document}x , y , zv .
using the absorbing property of \documentclass[12pt]{minimal } 
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				 \begin{document}$$p , f , g , h , r , s$$\end{document}p , f , g , h , r , s and \documentclass[12pt]{minimal } 
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				 \begin{document}$$t$$\end{document}t we obtain the following .
first , \documentclass[12pt]{minimal } 
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				 \begin{document}$$c=0$$\end{document}c=0 because \documentclass[12pt]{minimal } 
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				 \begin{document}$$p(0,0,0)=0$$\end{document}p(0,0,0)=0 from ( 2.4 ) .
then , we substitute \documentclass[12pt]{minimal } 
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				 \begin{document}$$y = z=0$$\end{document}y = z=0 in ( 2.4 ) and obtain \documentclass[12pt]{minimal } 
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				 \begin{document}$$f(x)=0$$\end{document}f(x)=0 .
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				 \begin{document}$$g(y)=h(z)=0$$\end{document}g(y)=h(z)=0 .
it remains \documentclass[12pt]{minimal } 
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				 \begin{document}$$p(x , y , z)=r(x , y)+s(x , z)+t(y , z)$$\end{document}p(x , y , z)=r(x , y)+s(x , z)+t(y , z ) .
we have \documentclass[12pt]{minimal } 
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				 \begin{document}$$t(y , z)=0$$\end{document}t(y , z)=0 , because \documentclass[12pt]{minimal } 
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				 \begin{document}$$0=p(0,y , z)$$\end{document}0=p(0,y , z ) .
analogously , we obtain \documentclass[12pt]{minimal } 
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				 \begin{document}$$r(x , y)=s(x , z)=0$$\end{document}r(x , y)=s(x , z)=0 .
whence , \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v is \documentclass[12pt]{minimal } 
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				 \begin{document}$$2$$\end{document}2-supernilpotent , by [ 2 , corollary 6.12 ] .
\documentclass[12pt]{minimal } 
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				 \begin{document}$$\square $ $ \end{document}
  ( cf . [ 5 , corollary 7.4 ] ) let \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a be a nilpotent malcev algebra with a malcev term \documentclass[12pt]{minimal } 
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				 \begin{document}$$d$$\end{document}d and let \documentclass[12pt]{minimal } 
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				 \begin{document}$$o\in a$$\end{document}oa .
then for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_1,a_2,b_1,b_2\in a$$\end{document}a1,a2,b1,b2a there exist \documentclass[12pt]{minimal } 
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				 \begin{document}$$x , y\in a$$\end{document}x , ya such that \documentclass[12pt]{minimal } 
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				 \begin{document}$$d(x , o , a_1)=b_1$$\end{document}d(x , o , a1)=b1 and \documentclass[12pt]{minimal } 
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				 \begin{document}$$d(a_2,o , y)=b_2$$\end{document}d(a2,o , y)=b2 .   the function \documentclass[12pt]{minimal } 
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				 \begin{document}$$x\mapsto d(x , o , a_1)$$\end{document}xd(x , o , a1 ) is bijective for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_1\in a$$\end{document}a1a , by [ 5 , corollary 7.4 ] .
hence , the equation \documentclass[12pt]{minimal } 
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				 \begin{document}$$d(x , o , a_1)=b_1$$\end{document}d(x , o , a1)=b1 has a unique solution for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_1,b_1\in a$$\end{document}a1,b1a .
we know that \documentclass[12pt]{minimal } 
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				 \begin{document}$$d(x , y , z):=d(z , y , x)$$\end{document}d(x , y , z):=d(z , y , x ) for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$x , y , z\in a$$\end{document}x , y , za is also a malcev term of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a .
therefore , \documentclass[12pt]{minimal } 
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				 \begin{document}$$y\mapsto d(y , o , a_2)$$\end{document}yd(y , o , a2 ) is bijective for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_2\in a$$\end{document}a2a by [ 5 , corollary 7.4 ] .
hence , \documentclass[12pt]{minimal } 
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				 \begin{document}$$y\mapsto d(a_2,o , y)$$\end{document}yd(a2,o , y ) is bijective for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_2\in a$$\end{document}a2a .
whence , the equation \documentclass[12pt]{minimal } 
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				 \begin{document}$$d(a_2,o , y)=b_2$$\end{document}d(a2,o , y)=b2 has a unique solution for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_2,b_2\in a$$\end{document}a2,b2a .
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				 \begin{document}$$\square $ $ \end{document}   ( cf .
[ 7 , theorem 1.2 ] ) every semigroup \documentclass[12pt]{minimal } 
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				 \begin{document}$$(g,+)$$\end{document}(g,+ ) such that the equations \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_1+x = b_1$$\end{document}a1+x = b1 and \documentclass[12pt]{minimal } 
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				 \begin{document}$$y+a_2=b_2$$\end{document}y+a2=b2 are solvable for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_1,a_2,b_1,b_2\in a$$\end{document}a1,a2,b1,b2a , is a group .   
\documentclass[12pt]{minimal } 
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				 \begin{document}$$\square $ $ \end{document}   for a malcev algebra \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a the following are equivalent:\documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a is \documentclass[12pt]{minimal } 
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				 \begin{document}$$2$$\end{document}2-supernilpotent \documentclass[12pt]{minimal } 
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				 \begin{document}$$(=0)$$\end{document}(=0)\documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a is polynomially equivalent to an expanded group \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v = ( a,+,-,0,f)$$\end{document}v=(a,+,-,0,f ) such that\documentclass[12pt]{minimal } 
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				 \begin{document}$$f$$\end{document}f is a set of at most binary absorbing operations on \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v , every absorbing operation in \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathsf{pol}_2(\mathbf v ) $ $ \end{document}pol2(v ) is distributive with respect to \documentclass[12pt]{minimal } 
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				 \begin{document}$$+$$\end{document}+ on both arguments , and\documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v is \documentclass[12pt]{minimal } 
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				 \begin{document}$$2$$\end{document}2-nilpotent .
\documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a is \documentclass[12pt]{minimal } 
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				 \begin{document}$$2$$\end{document}2-supernilpotent \documentclass[12pt]{minimal } 
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				 \begin{document}$$(=0)$$\end{document}(=0 ) \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a is polynomially equivalent to an expanded group \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v = ( a,+,-,0,f)$$\end{document}v=(a,+,-,0,f ) such that\documentclass[12pt]{minimal } 
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				 \begin{document}$$f$$\end{document}f is a set of at most binary absorbing operations on \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v , every absorbing operation in \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathsf{pol}_2(\mathbf v ) $ $ \end{document}pol2(v ) is distributive with respect to \documentclass[12pt]{minimal } 
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				 \begin{document}$$+$$\end{document}+ on both arguments , and\documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v is \documentclass[12pt]{minimal } 
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				 \begin{document}$$2$$\end{document}2-nilpotent .
\documentclass[12pt]{minimal } 
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				 \begin{document}$$f$$\end{document}f is a set of at most binary absorbing operations on \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v , every absorbing operation in \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathsf{pol}_2(\mathbf v ) $ $ \end{document}pol2(v ) is distributive with respect to \documentclass[12pt]{minimal } 
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				 \begin{document}$$+$$\end{document}+ on both arguments , and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v is \documentclass[12pt]{minimal } 
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				 \begin{document}$$2$$\end{document}2-nilpotent .
\documentclass[12pt]{minimal } 
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				 \begin{document}$$(1)\rightarrow ( 2)$$\end{document}(1)(2 ) using [ 2 , ( hc8 ) ] we obtain \documentclass[12pt]{minimal } 
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				 \begin{document}$$[1,]=0$$\end{document}[1,]=0 .
therefore , \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a is a 2-nilpotent malcev algebra .
let \documentclass[12pt]{minimal } 
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				 \begin{document}$$o\in a$$\end{document}oa and let \documentclass[12pt]{minimal } 
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				 \begin{document}$$d$$\end{document}d be a malcev term .
we define \documentclass[12pt]{minimal } 
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				 \begin{document}$$+:a^2\rightarrow a$$\end{document}+:a2a by\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } x+y:=d(x , o , y ) \end{aligned}$$\end{document}x+y:=d(x , o , y)for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$x , y\in a$$\end{document}x , ya . from proposition  3.1
we know that the equations \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_1+x = b_1$$\end{document}a1+x = b1 and \documentclass[12pt]{minimal } 
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				 \begin{document}$$y+a_2=b_2$$\end{document}y+a2=b2 are solvable for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_1,a_2,b_1,b_2\in a$$\end{document}a1,a2,b1,b2a .
let us show that \documentclass[12pt]{minimal } 
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we observe that the polynomial\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } p(x , y , z):=d(d(d(x , o , y),o , z),d(x , o , d(y , o , z)),o ) \end{aligned}$$\end{document}p(x , y , z):=d(d(d(x , o , y),o , z),d(x , o , d(y , o , z)),o)for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$x , y , z\in a$$\end{document}x , y , za is an absorbing polynomial at \documentclass[12pt]{minimal } 
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				 \begin{document}$$(o , o , o)$$\end{document}(o , o , o ) with value \documentclass[12pt]{minimal } 
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				 \begin{document}$$o$$\end{document}o .
therefore , \documentclass[12pt]{minimal } 
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				 \begin{document}$$(p(a , b , c),o)\in $ $ \end{document}(p(a , b , c),o) for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$a , b , c\in a$$\end{document}a , b , ca by [ 2 , lemma 6.9 ] . using the assumption \documentclass[12pt]{minimal } 
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				 \begin{document}$$=0$$\end{document}=0 , we obtain \documentclass[12pt]{minimal } 
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				 \begin{document}$$p(a , b , c)=o$$\end{document}p(a , b , c)=o .
equivalently,\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } d(d(d(a , o , b),o , c),d(a , o , d(b , o , c)),o)=o .
\end{aligned}$$\end{document}d(d(d(a , o , b),o , c),d(a , o , d(b , o , c)),o)=o.by [ 5 , corollary 7.4 ] we have that \documentclass[12pt]{minimal } 
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				 \begin{document}$$x\mapsto d(x , d(a , o , d(b , o , c)),o)$$\end{document}xd(x , d(a , o , d(b , o , c)),o ) is a bijective mapping of \documentclass[12pt]{minimal } 
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				 \begin{document}$$a$$\end{document}a .
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				 \begin{document}$$d(d(a , o , b),o , c)=d(a , o , d(b , o , c))$$\end{document}d(d(a , o , b),o , c)=d(a , o , d(b , o , c ) ) , because \documentclass[12pt]{minimal } 
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				 \begin{document}$$d(d(d(a , o , b),o , c),d(a , o , d(b , o , c)),o)=d(d(a , o , d(b , o , c)),d(a , o , d(b , o , c)),o)$$\end{document}d(d(d(a , o , b),o , c),d(a , o , d(b , o , c)),o)=d(d(a , o , d(b , o , c)),d(a , o , d(b , o , c)),o ) . using the recently introduced notation we have proved \documentclass[12pt]{minimal } 
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				 \begin{document}$$(a+b)+c = a+(b+c)$$\end{document}(a+b)+c = a+(b+c ) .
now using lemma  3.2 we obtain that \documentclass[12pt]{minimal } 
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				 \begin{document}$$+$$\end{document}+ is a group operation with the neutral element \documentclass[12pt]{minimal } 
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				 \begin{document}$$o$$\end{document}o .
we denote the inverse of an element \documentclass[12pt]{minimal } 
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				 \begin{document}$$a\in a$$\end{document}aa by \documentclass[12pt]{minimal } 
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				 \begin{document}$$-a$$\end{document}-a .
this is a polynomial operation given by\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } -x:=d(o , d(d(o , x , o),o , x),d(o , x , o ) ) \end{aligned}$$\end{document}-x:=d(o , d(d(o , x , o),o , x),d(o , x , o))for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$x\in a$$\end{document}xa , by [ 5 , lemma 7.3 ] .
we have proved that \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a is polynomially equivalent to an expanded group \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v with the group reduct \documentclass[12pt]{minimal } 
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				 \begin{document}$$(a,+,-,o)$$\end{document}(a,+,-,o ) .
now , we shall prove that \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a is polynomially equivalent to the expanded group \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf{v } : = ( a,+,-,o , f)$$\end{document}v:=(a,+,-,o , f ) , where \documentclass[12pt]{minimal } 
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				 \begin{document}$$f:=p_o(\mathbf{a } ) \cup cp(\mathbf{a } ) $ $ \end{document}f:=po(a)cp(a ) .
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				 \begin{document}$$\mathbf v $ $ \end{document}v are polynomials of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a . by [ 2 , ( hc3 ) ] , we know that \documentclass[12pt]{minimal } 
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				 \begin{document}$$[\,\underbrace{1,\ldots , 1}_n\,]=0$$\end{document}[1,  ,1n]=0 for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$n\ge 3$$\end{document}n3 . hence , \documentclass[12pt]{minimal } 
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				 \begin{document}$$f=0$$\end{document}f=0 for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$n$$\end{document}n - ary absorbing polynomial operations \documentclass[12pt]{minimal } 
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				 \begin{document}$$f$$\end{document}f of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a if \documentclass[12pt]{minimal } 
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				 \begin{document}$$n\ge 3$$\end{document}n3 , by [ 2 , corollary 6.12 ] .
hence , the set of all non - constant absorbing polynomial operations of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a is \documentclass[12pt]{minimal } 
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				 \begin{document}$$f:=p_o(\mathbf{a } ) \cup cp(\mathbf{a } ) $ $ \end{document}f:=po(a)cp(a ) . using lemma  2.2
we obtain that each polynomial of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a is also a polynomial of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf v $ $ \end{document}v .
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				 \begin{document}$$f\in cp(\mathbf{v } ) $ $ \end{document}fcp(v ) .
one can easily see that the polynomial \documentclass[12pt]{minimal } 
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				 \begin{document}$$q$$\end{document}q defined by\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } q(x , y , z):=f(x , y+z)-f(x , z)-f(x , y ) \end{aligned}$$\end{document}q(x , y , z):=f(x , y+z)-f(x , z)-f(x , y)for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$x , y , z\in a$$\end{document}x , y , za is absorbing at \documentclass[12pt]{minimal } 
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				 \begin{document}$$(o , o , o)$$\end{document}(o , o , o ) with value \documentclass[12pt]{minimal } 
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				 \begin{document}$$o$$\end{document}o . therefore ,
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				 \begin{document}$$(q(a , b , c),o)\in $ $ \end{document}(q(a , b , c),o) by [ 2 , corollary 6.9 ] for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$a , b , c\in a$$\end{document}a , b , ca .
hence , \documentclass[12pt]{minimal } 
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				 \begin{document}$$q(a , b , c)=o$$\end{document}q(a , b , c)=o or equivalently,\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } f(a , b+c)=f(a , b)+f(a , c ) \end{aligned}$$\end{document}f(a , b+c)=f(a , b)+f(a , c)for all \documentclass[12pt]{minimal } 
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				 \begin{document}$$a , b , c\in a$$\end{document}a , b , ca , by the assumption \documentclass[12pt]{minimal } 
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				 \begin{document}$$=0$$\end{document}=0 .
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				 \begin{document}$$(2)\rightarrow ( 1)$$\end{document}(2)(1 ) polynomially equivalent algebras have the same congruence lattice and the same commutator operations acting on the congruence lattice , by [ 2 , corollary 6.11 ] .
therefore , \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathbf a $ $ \end{document}a is \documentclass[12pt]{minimal } 
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				 \begin{document}$$2$$\end{document}2-supernilpotent by lemma  2.3 .
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				 \begin{document}$$\square $ $ \end{document}
   in this note we prove that a malcev algebra is 2-supernilpotent ( [ 1 , 1 , 1 ]  =   0 ) if and only if it is polynomially equivalent to a special expanded group . 
 this generalizes gumm s result that a malcev algebra is abelian if and only if it is polynomially equivalent to a module over a ring . 
Introduction
Special expanded groups
Malcev algebras