ACTA issues

Varieties of algebras with linear terms

George Hutchinson

Acta Sci. Math. (Szeged) 65:1-2(1999), 33-45

Abstract. Suppose $\cal V$ is a pointed variety of algebras of a type $\tau $, that is, $\cal V$ has a nullary operation $0$ such that each $A$ in $\cal V$ has $\{0_A\} $ as its smallest subalgebra. Each $n$-ary term $p(x_1,x_2,\ldots,x_n)$ of $\cal V$ determines a corresponding $n$-tuple of unary terms $\langle p_1,p_2,\ldots,p_n\rangle $, where $$p_j(x)=p(0,0,\ldots,0,x,0,\ldots,0)$$ equals $p$ evaluated with $x$ as $j$-th argument and $0$ elsewhere. If each $n$-ary term $p$ is uniquely determined by $\langle p_1,p_2,\ldots,p_n\rangle $ up to equivalence in $\cal V$, then $\cal V$ is said to have {\it linear terms}. The variety $R$-Mod of modules over a ring $R$ with unit has linear terms, since each $n$-ary term is equivalent to some linear combination $\Sigma_{j=1}^nr_jx_j$ which is determined by an $n$-tuple of coefficients in $R$. More generally, the variety of semimodules over a semiring $S$ with unit has linear terms. Pointed unary varieties also have linear terms. The properties of varieties with linear terms are studied.

AMS Subject Classification (1991): 08B99

Received April 15, 1997. (Registered under 2670/2009.)