Clones of term functions of lattices and Abelian groups
type: Item
instance of: scholarly article
title: Clones of term functions of lattices and Abelian groups
zbMATH Open document ID: 0566.08003
DOI: 10.1007/BF01236803
published in: Algebra Universalis
publication date: 1985-01-01T00:00:00Z
review text: 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 constructe…
reviewed by: Grigore Călugăreanu
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 08A02
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 20K01
zbMATH DE Number: 3902725
zbMATH Keywords: n-ary function
zbMATH Keywords: unary function
zbMATH Keywords: clone of term functions
zbMATH Keywords: polynomial functions
zbMATH Keywords: varieties of clones
zbMATH Keywords: hyperidentities
zbMATH Keywords: clone equation
zbMATH Keywords: Abelian p-group
zbMATH Keywords: endomorphism ring
author: Dietmar Schweigert
MaRDI profile type: Publication
cites work: Q3910579
cites work: Q4769119
cites work: Hyperidentities of lattices and semilattices
cites work: On semigroups of polynomial functions
cites work: On varieties of clones
cites work: On Prepolynomially Complete Algebras
cites work: Hyperidentities and hypervarieties
cites work: Über endliche, ordnungsaffinvollständige Verbände
description: scientific article; zbMATH DE number 3902725