Date: Tue, 27 Sep 1994 11:42:57 +0500 (GMT+4:00)
From: categories <cat-dist@mta.ca>
Subject: question

Date: Tue, 27 Sep 1994 11:35:46 +0100
From: Michael Wendt <wendt@math.muni.cz>

I met someone at a conference recently who wanted me to post something to
the Cats bulletin board.

Forwarded message:

Ewa Graczynska of Wroclaw, Poland does research on hyperidentities and
solid varieties of algebras and would like to know if there is a connection
with lambda-calculus.

*
* In pseudo-latex code:
*
* Definition: Fix a type \tau =(n_i)_{i\in I}, n_i>0 for all i\in I and
* operation symbols (f_i)_{i\in I}, where f_i is n_i-ary. Let W_\tau(X) be the
* set of all terms of type \tau over the alphabet X=\{ x_1, x_2, ... \}. Terms
* W_\tau(X_n) with X_n =\{ x_1, x_2,..., x_n\} are called n-ary. Moreover, let
* Alg(\tau) be the class of all algebras of type \tau. Then a mapping
*
*              \sigma: \{ f_i: i\in I \} --------> W_\tau (X)
*
* which assigns to every n_i-ary operation symbol, f_i, an n_i-ary term will
* be called a HYPERSUBSTITUTION of type \tau. Hypersubstitutions can be uniquely* extended to terms:
*
*              \hat{\sigma}: W_\tau(X) --------> W_\tau(X)
*
* is defined inductively on the complexity of terms from W_\tau(X) in a
* natural way:
* (i) \hat{\sigma}[x]:=x for any variable x in the alphabet X;
* (ii) \hat{\sigma}[f_i(t_1,...t_{n_i})]:= {\sigma(f_i)}^{W_\tau(X)} (\hat{
* \sigma}[t_1],..., \hat{\sigma}[t_t_{n_i}]), where {\sigma(f_i)}^{W_\tau(X)}
* denotes the term operation induced by \sigma(f_i) on the term algebra on the
* universe W_\tau(X).
*
* By Hyp(\tau), we denote the set  of all hypersubstitutions of type \tau.
*
* An equation p=q of type \tau (a pair of terms of W_\tau(X)) is called a
* HYPERIDENTITY of type \tau in an algebra A\in Alg(\tau) iff
* \hat{\sigma}(p) = \hat{\sigma}(q) is an identity of A for every
* \hat{\sigma}\inHyp(\tau).
*
* SOLID varieties of algebras are varieties of algebras of a given type
* \tau in which all identities are hyperidentities as well. (end definition)
*
* References:
* [1] E. Graczynska, D. Swiegert; "Hypervarieties of a Given Type;" Algebra
* Universalis 27 (1990), 305-318.
*
* [2] W. Taylor; "Hyperidentities and Hypervarieties," Aequationes math. 23
* (1981), 30-49.
*

She would like to know if there are connections between hyperidentities
and lambda-calculus. She is not on the categories bulletin board. Please
don't  send a response to me. E-mail her directly at
WSIOPOLE@mvax.ci.pwr.wroc.pl