Clones of term functions of lattices and Abelian groups
Let H be a set of functions on A. The clone \({\mathbb{H}}=(H;*,\xi,\tau,\Delta,e^ 2_ 1)\) is an algebra of type (2,1,1,1,0) where the operations are defined in the following way: 1) \((f*g)(x_ 1,...,x_ m,x_{m+1},....,x_{m+n-1})=f(g(x_ 1,...,x_ m),x_{m+1},...,x_{m+n-1})\) for an n-ary function f and an m-ary function g. 2) \((\xi f)(x_ 1,...,x_ n)=f(x_ 2,...,x_ n,x_ 1)\) for an n-ary function f, \(n>1\), and \((\xi f)(x_ 1)=f(x_ 1)\) for a unary function f. 3) \((\tau f)(x_ 1,...,x_ n)=f(x_ 2,x_ 1,x_ 3,...,x_ n)\) and identical on unary functions. 4) \((\Delta f)(x_ 1,...,x_{n-1})=f(x_ 1,x_ 1,...,x_{n-1})\) and identical on unary functions; 5) \(e^ 2_ 1(x_ 1,x_ 2)=x_ 1\). The clone of all functions on A is denoted by F(A). Let \({\mathbb{A}}=(A,\Omega)\) be an algebra. The subclone T(\({\mathbb{A}})\) of F(A) generated by the operations of \({\mathbb{A}}\) is called the clone of term functions of \({\mathbb{A}}\). The clone P(\({\mathbb{A}})\) of polynomial functions of \({\mathbb{A}}\) is the subclone of F(A) generated by the constants \(c_ a\), \(c_ a(x)=a\), \(a\in A\), and the operations of \({\mathbb{A}}.\) For an n-ary function \(\phi\in {\mathbb{H}}\) we can construct the function \(\phi *\phi (x_ 1,...,x_{2n-1})=\phi (\phi (x_ 1,...,x_ n)x_{n+1},...,x_{2n-1})\) and by identifying \(x_{n+1}=x_ 2,...,x_{2n-1}=x_ n\) by the use of 2), 3), 4) we obtain the function \(\phi \circ \phi (x_ 1,...,x_ n)=\phi (\phi (x_ 1,...,x_ n),x_ 2,...,x_ n)\). Iterating we define \(\phi^ k\) for every \(k\in {\mathbb{N}}.\) In a previous paper the author has shown that to a given variety V of algebras \({\mathbb{A}}\) there corresponds a variety \({\mathbb{T}}(V)\) of clones. Equations of the variety \({\mathbb{T}}(V)\) hold for every clone T(\({\mathbb{A}})\) of term functions of \({\mathbb{A}}\) whenever \({\mathbb{A}}\in V\). For the variety V these equations of varieties of clones are sets of hyperidentities. In this paper the author is interested only in equations constructed by the composition ''\(\circ ''\) defined above. The results obtained are of the following kind: Theorem 1.5. The clone equation \(\phi^ 3=\phi^ 5\) holds for the variety \({\mathbb{T}}(N_ 5)\) which is generated by the clone of term functions of the lattice \(N_ 5\). The clone equation \(\phi^ 3=\phi^ 9\) holds for the variety \({\mathbb{T}}(M_ 3)\) which is generated by the clone of term functions of the lattice \(M_ 3\). Theorem 3.1. If \({\mathbb{G}}\) is a finite subdirectly irreducible Abelian group then the equation \(\phi^ n=\phi^{n+p^ n(p-1)}\) holds for \({\mathbb{T}}({\mathbb{G}})\) with \(| {\mathbb{G}}| =p^ n\), p a prime number. Theorem 3.4. Let \(({\mathbb{G}},+)\) be a finite elementary Abelian p-group and let End(\({\mathbb{G}})\) be the endomorphism ring of \({\mathbb{G}}\). The equation \(\phi^ n=\phi\) for some \(n\in {\mathbb{N}}\), \(n>1\), holds for End(\({\mathbb{G}})\) if and only if \(| {\mathbb{G}}| =p\).
scientific article; zbMATH DE number 3902725