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\centerline{\large \bf On Assertional Equivalence of Quasivarieties}

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\centerline{W. J. Blok}
\centerline{\it University of Illinois at Chicago, USA}
\centerline{e-mail {\tt wjb@uic.edu}}

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\centerline{J. G. Raftery}
\centerline{\it University of Natal, South Africa}
\centerline{e-mail {\tt raftery@nu.ac.za}}
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For a given algebraic language, a `translation' means a finite family 
$T$ of
pairs of unary terms $\langle d, e\rangle$; 
it is intended to translate terms $t$ into
equations $d(t) = e(t)$, and similarly derived (logical) rules 
$G \vdash t$ into
quasi-identities.

For a given quasivariety 
$\mathcal K$ of the same language, the $T$-assertional logic of
$\mathcal K$ 
is the relation from sets of terms to terms consisting of all pairs 
$\langle G, t\rangle$
for which $\mathcal K$ validates 
the quasi-identity that translates the formal
expression $G\vdash t$.  
This relation is indeed a logic so it induces on each
algebra of K a system of logical filters, i.e. subsets closed under
suitable axioms and inference rules determined by 
$\mathcal K$ and $T$.

The obvious example is that the classical propositional calculus arises
from Boolean algebras (BA) in this way using 
$\langle d, e\rangle = \langle x, 1\rangle \ldots$ but it can
be got in exactly the same way from the larger variety of dually
pseudocomplemented distributive lattices (DPDL).  Some well known
properties of DPDL can be reinterpreted as consequences of this fact.

Whereas BA is a variety of logic (in a sense that is now understood
precisely), DPDL is not \ldots 
but it belongs to a larger class that we shall
introduce and discuss, called $T$-semiregular quasivarieties.  They still
admit quite a smooth theory of correspondence between logical filters and
relative congruences on algebras.

When two different quasivarieties (like BA and DPDL) have the same
$T$-assertional logic we call them 
$T$-assertionally equivalent.  The talk
reports an attempt to study this equivalence relation as generally as
possible.

A quasivariety that is $T$-assertionally equivalent to the variety that it
generates is called $T$-assertionally conservative (this obviously includes
all varieties).  Several (but not all) quasivarieties from linear logic
have this feature (and are not varieties); BCK- and BCI-algebras (in
Iseki's sense) are among them.  We can give quite general explanations for
their relatively congruence modularity (with a reasonable commutator
theory), and to separate the relatively congruence distributive ones from
the others by a pure Mal'cev condition.  This contrasts with the following
known facts: A relatively point regular quasivariety need not generally be
relatively modular, and its relative distributivity is not characterized by
a pure Mal'cev condition.  And a variety of logic need satisfy no
nontrivial congruence identity.

The theory of $T$-semiregular and 
$T$-assertionally conservative quasivarieties
also includes a generalization of the notion of a subtractive variety (in
the sense of Agliano and Ursini), with its desirable filter-generation
properties.  Subtractive varieties have a definable constant 
$1$ and $\langle x, 1\rangle$ is
the appropriate translation.  The generalized notion can be applied
usefully, e.g., to Sugihara algebras (which have no definable constant).




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{\bf  Presented by James G. Raftery}
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