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\centerline{\large \bf Techniques in finite-basis problems}

\bigskip

\centerline{Kirby A. Baker}
\centerline{\it University of California, Los Angeles, USA}
\centerline{e-mail {\tt baker@math.ucla.edu}}

\bigskip
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%%
\par
The structures of algebraic logic are typically congruence
distributive, or more generally, congruence
meet-semidistributive.  Congruence distributivity supports
strong theorems on the existence of finite equational bases
of varieties.  To adapt these theorems and their proof techniques
to the more general setting has presented challenges that are
being met through the work of Willard and others.  The talk
will discuss recent contributions to such techniques.

\newpage
\centerline{\large \bf Varieties with Equationally Definable Order
Relations}

\bigskip

\centerline{Joel Berman}
\centerline{\it University of Illinois at Chicago, Chicago, IL, U.S.A.}
\centerline{e-mail {\tt jberman@uic.edu}}


\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\par
%   Text of the abstract
We consider varieties of algebras for which an order relation
on the universe of each algebra in the variety is uniformly definable
by conjunctions of equations.
Which varieties have nontrivial equationally definable order relations 
and which classes of ordered sets  arise from such varieties?  
We investigate free algebras in these
varieties and those algebras in the variety freely generated
by individual ordered sets.  
We also consider how, given a class of ordered sets, there
is a uniform way to associate an algebra with each ordered set
in the class so that the variety generated by these algebras
has an equationally definable order relation.  
We investigate ways of constructing such algebras 
from some specific classes
of ordered sets and discuss some of the algebraic properties
of the resulting varieties. 

\newpage
\centerline{\large \bf Discriminator Varieties with Group Stalks}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Dejan Deli\'c}
\centerline{\it Ryerson University, Canada}
\centerline{e-mail {\tt ddelic@acs.ryerson.ca}}

%%% in case of several authors:
%\smallskip
%\centerline{Author 2}
%\centerline{\it Institution Name, Country}
%\centerline{e-mail {\tt Author 2}}
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\bigskip
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\par
%   Text of the abstract
In this talk we investigate conditions that need to be imposed on
a locally finite universally axiomatized class of groups so that
it generates a decidable locally finite discriminator variety in a
finite language. In particular, we will discuss the case when the
locally finite class of groups is a universal subclass of
$\mathcal{A}_p\mathcal{A}_q$ ($p\neq q$), the variety of Abelian
$p$-groups by Abelian $q$-groups. We will make an attempt to cast
the light on the intrinsically model-theoretic character of these
conditions, which present finite perturbations of the standard
model-theoretic property of homogeneity.

\newpage
\centerline{\large \bf Generative complexity in algebra}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Pawe\l \ M.~Idziak}
\centerline{\it Jagiellonian University, Krak\'ow, Poland}
\centerline{e-mail {\tt idziak@ii.uj.edu.pl}}

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\par
%   Text of the abstract
The generative complexity of a class of models is the function that counts,
up to isomorphism, the number of at most $k$--generated models in the class.
The infinite counterpart of
the problem of counting non-isomorphic models
is widely studied in Model Theory, and is one of the fundamental topics
for Shelah's classification theory and for stability theory.
Note that in the infinite realm (and for a countable language)
being $\kappa$-generated and having $\kappa$ elements are the same.

However in the finite setting it is enormously hard to count models,
especially algebras,
with a given number of elements.
One possible way to overcome these difficulties is to count
$k$-generated models instead of $k$-element ones.
This is a more tractable problem when the classes are varieties of algebras.
Note however that these numbers are the same for purely relational languages.

Two main results in the area are:
\\
{\bf [P.Idziak, R.McKenzie, M.Valeriote]}
{\em a full characterization of locally finite varieties
with polynomial generative complexity
as varietal products of an affine variety over a finite ring of finite representation type
and several varieties that are matrix powers of finite pointed $G$--sets},
\\
{\bf [J.Berman, P.Idziak]}
{\em a full characterization of finitely generated congruence modular varieties
%(or more generally varieties that omit type {\bf 1})
with singly exponential generative complexity}.

The conditions for locally finite varieties with polynomial generative complexity
are very simple and easily stated.
The conditions involved in the second characterization,
are more complicated,
although they also have a natural algebraic meaning.
In both cases we know that the bound for the number of algebras
implies a very transparent and manageable structure.
For example, when specializing our results to groups we get the following:
\emph{
\\
\indent$\bullet$
every finitely generated variety of groups
has at most doubly exponential generative complexity,
\\
\indent$\bullet$
it has singly exponential generative complexity
if and only if it is nilpotent,
\\
\indent$\bullet$
it has
polynomial generative complexity
if and only if it is Abelian,
\\
\em while for commutative rings with unit our characterization reduces to: \em
\\
\indent$\bullet$
every finitely generated variety of rings has at most doubly
exponential generative complexity,
\\
\indent$\bullet$
a finite commutative ring with unit generates a variety with singly exponential generative complexity
iff its Jacobson radical squares to $0$,
\\
\indent$\bullet$
no nontrivial variety of rings with unit has
polynomial generative complexity.
}

\newpage
\centerline{\large \bf On categorical equivalence of finite arithmetical
algebras} 

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Kalle Kaarli}
\centerline{\it University of Tartu, Estonia}
\centerline{e-mail {\tt kaarli@math.ut.ee}}

\bigskip
%%
\par
%   Text of the abstract
Two algebras $\bf A$ and $\bf B$ are called {\it categorically equivalent}
if there is a functor $F$ which is an equivalence from the category
$V({\bf A})$ to $V({\bf B})$ such that $F({\bf A})=\bf B$. C.~Bergman %[1]
completely solved the categorical equivalence problem for finite algebras
with majority term. As a special case, he proved that the categorical
equivalence class 
of a finite algebra $\bf A$ which has no proper subalgebras and generates an
arithmetical variety, is fully determined by the full subcategory 
$\bf Q(A)$ of $V({\bf A})$ whose objects are the quotient algebras
of $\bf A$. The present work addresses the question what is the formal
structure of the categories $\bf Q(A)$ which appear in this situation. 
So far we have mainly studied algebras satisfying no isomorphic quotients
condition {\bf (NIQ)}. The latter means that the quotients ${\bf A}/\rho$
and ${\bf A}/\sigma$ are isomorphic only in case $\rho=\sigma$.
If an algebra $\bf A$ satisfies {\bf (NIQ)} then, given arbitrary
$\rho,\sigma\in{\rm Con}\bf A$, $\rho\leq\sigma$, every 
$\alpha\in{\rm Aut}{\bf A}/\rho$ induces 
$\pi_{\rho\sigma}(\alpha)\in{\rm Aut}{\bf A}/\sigma$
and the mapping $\pi_{\rho\sigma}$ is a group homomorphism.
The system of groups ${\rm Aut}{\bf A}/\rho$ and homomorphisms
$\pi_{\rho\sigma}$ is a natural example of what we call a {\it group
scheme}. We have proved that two finite algebras satisfying {\bf (NIQ)},
having no proper subalgebras, and generating arithmetical varieties are
categorically equivalent iff their group schemes are isomorphic.
Our main result, however, gives a complete characterization of group
schemes appearing in this situation. 
%\medskip
%{\bf References} 1. C. Bergman, {\it Categorical equivalence of algebras
%with majority term}. Algebra Universalis {\bf 40} (1998), 149--175.

\newpage
\centerline{\large \bf About $n$-permutable varieties
 and congruence identities}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Paolo Lipparini}
\centerline{\it Universit\`a di Tor Vergota, Roma, Italia}
\centerline{e-mail {\tt lipparin@axp.mat.uniroma2.it}}

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\par
%   Text of the abstract

We proved that for every $n\geq 2$ there is a non-trivial
 lattice
 identity holding in congruence lattices of algebras
 belonging
to any $n$-permutable variety (partial results towards
 this direction were
 obtained by G. Cz\'edli, A. Day, and D.
 Hobby and R. McKenzie).

Recall that two equivalence relations $\alpha,\beta $
 $n$-{\it permute}  if and only if $\alpha \circ \beta
 \circ
\alpha \circ \dots
=\beta \circ \alpha \circ \beta \circ \dots$ ($n$ factors
 on each side); an algebra is $n${\it -permutable} if
 and only
 if any two congruences $n$-permute, and a variety is
 $n${\it -permutable} if and only if all algebras in the
 variety are $n$-permutable.

$n$-permutability is an immediate generalization of
 permutability (that is, $2$-permutability); the
 well-known
 characterization by Mal'cev of permutable varieties can
 be safely considered as the beginning of modern
 Universal algebra. After a while, it turned out that
the notion of an $n$-permutable varieties is not just an
 auxiliary concept; rather, it has a key role in the
 classification of varieties
(just to state an example, there is a clear
 characterization of $n$-permutable locally finite
 varieties via tame congruence theory: they are exactly
 those varieties omitting types {\bf 1,4,5}).

Our original proof that $n$-permutable varieties satisfy
 congruence identities used the (non modular) commutator;
 but you have no need to know commutator theory, in order
 to listen to this communication, since we present here a
commutator-free proof. Indeed, in the particular cases
 $n=4$
and $n=5$, the proof we present is extremely short,
 elementary and simple
(cases $n=2$ and $n=3$ are well-known, since
$3$-permutability implies modularity).

If time allows, we shall discuss possible generalizations
of the methods to a class of varieties larger than
$n$-permutable ones.

\newpage
\centerline{\large \bf Finite axiomatizability of finitely generated classes}

\bigskip

\centerline{Ralph McKenzie}
\centerline{\it Vanderbilt University, Nashville, TN, USA}
\centerline{e-mail {\tt mckenzie@math.vanderbilt.edu}}
%%%

\bigskip
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%%
\par
%   Text of the abstract
We will review what is known about the following problems.
(1) Describe the class of finite algebras with finitely axiomatizable
equational theory.  (2) Describe the class of finite algebras with
finitely axiomatizable quasi-equational theory.

\newpage
\centerline{\large \bf Term equivalence of finite groups}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Keith A.\ Kearnes}
\centerline{\it University of Colorado, Boulder CO, USA}
\centerline{e-mail {\tt kearnes@euclid.colorado.edu}}

%%% in case of several authors:
\smallskip
\centerline{\'Agnes Szendrei}
\centerline{\it Bolyai Institute, University of Szeged, Hungary}
\centerline{e-mail {\tt a.szendrei@math.u-szeged.hu}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\par
%   Text of the abstract
Is a finite group determined up to isomorphism
by the subgroup lattices of its finite powers?
In this talk we refute the long-standing conjecture that the
answer to the question is affirmative. The 
counterexample consists of a pair
of non-isomorphic finite groups that are
term equivalent. 
Starting from this example, we will discuss
term equivalence in special classes of finite groups.

\bigskip

%%% in case of several authors:
{\bf  Presented by \'Agnes Szendrei}
%%%

\newpage
\centerline{\large \bf An equational approach to Glivenko like
theorems}

\bigskip

\centerline{R. Cignoli}

\centerline{\it Universidad de Buenos Aires and CONICET,
Argentina}

\centerline{e-mail {\tt cignoli@dm.uba.ar}}


\smallskip

\centerline{A. Torrens}

\centerline{\it Universidad de Barcelona, Spain}

\centerline{e-mail {\tt torrens@mat.ub.es}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\par

Let $\mathbf{A} = \langle A, \rightarrow, \top, \bot\rangle$ be a
bounded BCK algebra. For each $x \in A$, define $\neg x = x
\rightarrow \bot$, and let $Reg(\mathbf{A}) = \{x \in A : \neg
\neg x = x\}$.  $Reg(\mathbf{A})$ is a subuniverse of $A$.
Necessary and sufficient conditions are given in order that
(1)$\neg\neg :Reg(\mathbf{A}) \rightarrow A$ be a homomorphism,
and (2) $Reg(\mathbf{A})$ be a Boolean algebra. These conditions
are extended to expansions of bounded BCK algebras by means of
implicitly defined compatible operations, like bounded pocrims and
hoops, bounded BCK algebras endowed with a semilattice structure,
residuated lattices, Heyting algebras, BL-algebras.
 From the above results Glivenko like theorems are obtained for
 several  extensions of BCK logic.

\bigskip

 {\bf  Presented by R. Cignoli}

\newpage
\centerline{\large\bf Set-Theoretic Domains.}
\centerline{\large\bf Towards the Algebraization of Set Theory.}
\bigskip
\centerline{Janusz Czelakowski}
\centerline{\it Institute of Mathematics and Computer Science}
\centerline{\it Opole University, Oleska 48, Opole, Poland}
\centerline{e-mail: {\tt jczel@math.uni.opole.pl}}

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
%   Text of the abstract
Models of set theory are traditionally viewed as pairs $(A,e)$, where $e$ is 
a binary relation on a set $A$. In the algebraic approach they are  
quadruples $\boldsymbol{A}=(A,\leq,\&,\boldsymbol{0})$, 
called {\it set-theoretic domains}, where: 
(A1). $A$ is possibly a class partially ordered by $\leq$ and
$\boldsymbol{0}$ is the least element in the sense of $\leq$,
(A2). $\&$ is a binary operation on $A$ such that for any $a,b,c\in A$,
\vskip -5pt\hskip 20mm
(A2a)\quad $a\&b\leq c$  iff  $a\leq c$ and $\boldsymbol{0}\&b\leq c$,
\vskip -5pt\hskip 20mm
(A2b)\quad $\boldsymbol{0}\&c\leq a\&b$  implies  $a\&c=a$ or $c=b$,
\vskip -5pt\hskip 20mm
(A2c)\quad $\boldsymbol{0}\&a\not=\boldsymbol{0}\land\boldsymbol{0}\&a\not=a$.

It is also assumed that for each each $a\in A$, $E(a):=\{x\in A:a\&x=a\}$ 
is a set. $E(a)$ is called the {\it extension of} $a$.
Furthermore $A$ is assumed to satisfy: Infinity, the properties of 
Replacement, Union and Power for each $a\in A$ such that $E(a)$ is 
$\leq$-directed, and the following Weak Induction Principle:

For any property $P$ of elements of $A$,
$P(\boldsymbol{0})\land(\forall\,a\in A)(P(a)\to(\forall\,b\in A)P(a\&b))\land
(\forall\,a\in A)$ 
[$E(a)$ is $\leq$-directed $\land(\forall\,d\in E(a))P(d)\to 
P(\sup(E(a)))]\to(\forall\,a\in A)P(a)$.

Intuitively, the elements of $A$ are sets in the sense of $\boldsymbol{A}$.
$\leq$ is the inclusion relation in the sense of $\boldsymbol{A}$.
$\boldsymbol{0}$ is the empty $\boldsymbol{A}$-set. $\&$ is the operation
of adjoining an $\boldsymbol{A}$-set to a given $\boldsymbol{A}$-set
as a new element. $\&$ thus corresponds to the set-theoretic operation
which, when performed on sets $X$ and $Y$, yields the set $X\&Y:=X\cup\{Y\}$.
The $\in$-relation is then defined in terms of $\&$ by the condition:
$Y\in X$ if and only if $X\&Y=X$.


{\bf Theorem 1}.
{\it Let $\boldsymbol{A}=(A,\leq,\&,\boldsymbol{0})$ be a domain. 
Define the relation $\in_{\boldsymbol{A}}$ on $A$ by the stipulation: 
$b\in_{\boldsymbol{A}}a$ iff $a\&b=a$, for all $a,b\in A$ and put: 
$\boldsymbol{A}^0:=(A,\in_{\boldsymbol{A}})$. Then $\boldsymbol{A}^0$ is a model 
of the unrestricted {\rm ZF$_0$} $($= {\rm ZF} without the Axiom of Regularity$)$. 
Moreover $\boldsymbol{0}$ is the 
empty set in the sense of $\boldsymbol{A}^0$, $\leq$ coincides with the 
inclusion relation in the sense of $\boldsymbol{A}^0$, and, for any 
$a,b \in A$, $a\&b=a\cup_{\boldsymbol{A}^0}\{b\}_{\boldsymbol{A}^0}$.}

The converse of Theorem 1 is also true:

{\bf Theorem 2}. 
{\it Let $\boldsymbol{A}=(A,e)$ be a model of the unrestricted 
{\rm ZF$_0$}. Define: 
$\leq:=$ the inclusion relation in the sense of the model 
$\boldsymbol{A}$, i.e., for any $a,b\in A$: $a\leq b$ iff 
$(\forall\,c\in A)$ $(c\,e\,a$ implies $c\,e\,b)$; 
$\boldsymbol{0}:=$ the empty set in the sense of $\boldsymbol{A}$;
$a\&b:=$ the $\boldsymbol{A}$-union $a\cup_{\boldsymbol{A}}\{b\}_{\boldsymbol{A}}$, 
where $\{b\}_{\boldsymbol{A}}$ is the $\boldsymbol{A}$-singleton of $b$ 
and $\cup_{\boldsymbol{A}}$ is the $\boldsymbol{A}$-union of two 
$\boldsymbol{A}$-sets. Then the system 
$\boldsymbol{A}^\star:=(A,\leq,\&,\boldsymbol{0})$ is a domain.}

\newpage
\centerline{\large \bf On generalizations of measurable functions}

\bigskip

\centerline{Miklos Ferenczi} 
\centerline{\it TUB Department of Algebra, Hungary} 
\centerline{ferenczi@math.bme.hu}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%

The concept of a measurable function 
is of central importance in measure theory,
probability theory, etc.. Let $p$ be a transformation from a unit $X$ of a
Boolean set algebra $\mathcal{B}$ into a unit $V$ of another Boolean set
algebra $\mathcal{A}.$ It is known that for\emph{\ any function} $%
p:X\rightarrow V$ the pre-imagine (original) $\widetilde{p}$ associated with 
$p$ is a homomorphism from $\mathcal{A}$ into the Boolean power set algebra
with unit $X.$ It is easy to check that measurableness of $p$ means that $%
\widetilde{p}$ transforms,\textit{\ in additional,} into the universe $B$ of 
$\mathcal{B}$. We generalize the concept measurableness to \textit{cylindric
algebras}.\newline
Let us consider the direct generalizations of the Boolean case, the
canonical representations of cylindric algebras\emph{.} Let $\mathcal{A}$
and $\mathcal{B}$ the canonical representations of the cylindric algebras $%
\mathcal{A}^{\prime }$and $\mathcal{B}^{\prime }$ and let $V,X$ and $A,B$
the units and the universes of $\mathcal{A}$ and $\mathcal{B}$ respectively.
Let $\overset{\varkappa }{\thickapprox }$ denote the equivalence relation
defined on $X:I$ $\overset{\varkappa }{\thickapprox }J$ ($I,J\in X)$ if and
only if $I\cap Cl_{\varkappa }\mathcal{B}=J\cap Cl_{\varkappa }\mathcal{B}$
where $Cl_{\varkappa }\mathcal{B}$ is the set$\left\{ b\in B:c_{\varkappa
}b=b\right\} ,\varkappa <\alpha .$ Let $\overset{\varkappa }{\thicksim }$
denote the similar equivalence relation defined on $V.$\newline
Let $p:X\rightarrow V$ be a measurable function with respect to the Boolean
parts of $\mathcal{A}$ and $\mathcal{B}$ in the usual sense, i.e. let $%
\widetilde{p}$ transform into $B.$ Then the following theorem is true:

$\widetilde{p}\in Hom(\mathcal{A},\mathcal{B)}$ if and only if the
conditions (i)-(iii) hold for every $\lambda ,\varkappa <\alpha $:\newline
(i) if $I\overset{\varkappa }{\thickapprox }J$ then $pI\overset{\varkappa }{%
\thicksim }pJ$\newline
(ii) if $K\overset{\varkappa }{\thicksim }pJ,$ $K\in a$ for some $a\in A$
then there is an $I\in X$ such that $I\overset{\varkappa }{\thickapprox }J$
and $pI\in a$\newline
(iii) $I\in D_{\varkappa \lambda }^{\mathcal{B}}$ if and only if $pI\in
D_{\varkappa \lambda }^{\mathcal{A}}$ (where $D_{\varkappa \lambda }^{{}}$'s
denote the diagonals)
\newline
We can prove that the following condition (ii)' is sufficient for (ii):%
\newline
(ii)' if $K\in V$ is such that there are $J\in X$ and $\varkappa <\alpha $
so that $K\overset{\varkappa }{\thicksim }pJ$ then there is an $I\in X$ for
which $J\overset{\varkappa }{\thickapprox }I$ and $pI=K.$\newline
The meaning of (i), (ii)' and (iii) together is that $p$ is a bijection on
any equivalence class w.r.t. $\overset{\varkappa }{\thickapprox }$ to some
equivalence class w.r.t. $\overset{\varkappa }{\thicksim }.$ In contrary to
the Boolean case, for any $p:X\rightarrow V,\;\;\widetilde{p}$ is\emph{\ not 
}a homomorphism \ from $\mathcal{A}$ into the cylindric canonical embedding
algebra $\mathcal{E}m\mathcal{B}^{\prime }$ (into a complete algebra)
extending the algebra $\mathcal{B},$ but it is easy to check that (i), (ii)'
and (iii) imply that $\widetilde{p}$ is already a homomorphism.\newline
There are other \emph{differences }between the Boolean case and the
cylindrical case, too. There are many classes of representant cylindric set
algebras: $Crs\cap CA,$ $Gws,$ $Gs,$ etc. Further, the function $p$ may be
generated in terms of other functions defined on the bases or on the
subbases of the cylindric set algebras. The concept of measurable function
can be discussed for these algebras and functions, too. The point is that
the generating function restricted to certain equivalence classes must be a
bijection to certain equivalence classes.

\newpage
\centerline{\large \bf Fully adequate Gentzen systems in Abstract Algebraic Logic}

\bigskip

\centerline{Josep M. Font}
\centerline{\it University of Barcelona, Spain}
\centerline{{\tt font@mat.ub.es}}

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\par
%   Text of the abstract

\begingroup

\noindent\textbf{Definition [1].}
A Gentzen system $G$ is \textbf{\textit{fully adequate}} for a deductive system $\boldsymbol{\mathcal{S}}$ when a
generalized matrix is a model of
$G$ [or:\ a model of $G$ without theorems, in the case where $\boldsymbol{\mathcal{S}}$ has no theorems] if and only
if it is a full generalized model of $\boldsymbol{\mathcal{S}}$.

In joint work with \textsc{Ramon Jansana} (Barcelona) and \textsc{Don Pigozzi} (Iowa) we have
continued the study of full adequacy in the framework of Abstract Algebraic Logic. This notion
turns out to be quite interesting in that it relates several rather different parts of the
field. From the model-theoretic side, we have:

\noindent\textbf{Theorem [3].}
A deductive system has a fully adequate Gentzen system if and only if
the class of its full generalized models is closed under the constructions of substructures and
reduced products.

Some relations have arisen with different versions of the Deduction-Detachment Theorem (DDT):

\noindent\textbf{Theorem [1].}
All selfextensional deductive systems either with conjunction or with the uniterm DDT do
have a fully adequate Gentzen system.

We have also considered the following iterated (and weaker) version of the DDT:

\noindent\textbf{Definition [2].}
A deductive system $\boldsymbol{\mathcal{S}}$ has the \textbf{\textit{Graded
Deduction-Detachment Theorem}} (GDDT) when for each $n<\omega$ there is a set of formulas
$\varDelta_n(x_0,\dots,x_{n-1},y)$ such that for all formulas
$\varphi_0,\dots,\varphi_{n-1},\psi$ the following holds:
\[
\varphi_0,\dots,\varphi_{n-1} \vdash_{\!\boldsymbol{\mathcal{S}}} \psi\quad\iff\quad
\vdash_{\!\boldsymbol{\mathcal{S}}} \varDelta_n(\varphi_0,\dots,\varphi_{n-1},\psi)\,.
\]
We have weakened and generalized this property by \textit{adding parameters} to the
sets
$\varDelta_n$ and by \textit{relativizing the deducibility}
$\vdash_{\!\boldsymbol{\mathcal{S}}}$ to the so-called \textit{Leibniz theories} of
$\boldsymbol{\mathcal{S}}$. After some elaborated results on the GDDT for protoalgebraic
logics we finally obtain:

\noindent\textbf{Theorem [2].}
A weakly algebraizable deductive system $\boldsymbol{\mathcal{S}}$ has a fully adequate Gentzen system
if and only if it satisfies the multiterm DDT.
\bigskip

\noindent\textbf{\large References}
\small
\begin{enumerate}
\item 
{\sc Font, J.~M., and Jansana, R.}
\newblock {\em A general algebraic semantics for sentential logics}, vol.~7 of
  {\em Lecture Notes in Logic}.
\newblock Springer-Verlag, 1996.
\newblock 135 pp. Presently distributed by the Association for Symbolic Logic.
\item 
{\sc Font, J.~M., Jansana, R., and Pigozzi, D.}
\newblock Fully adequate {G}entzen systems and the deduction theorem.
\newblock {\em Reports on Mathematical Logic 35\/} (2001), 115--165.
\item 
{\sc Font, J.~M., Jansana, R., and Pigozzi, D.}
\newblock Fully adequate {G}entzen systems and closure properties of the class
  of full models.
\newblock Manuscript, to appear.
\end{enumerate}

\endgroup

\newpage
\centerline{\large \bf MV-algebras and multisets}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Daniele Mundici}
\centerline{\it Dept. of Computer Science, University of Milan,
via Comelico 39-41, 20135 Milan, Italy}
\centerline{e-mail:  {\tt mundici@mailserver.unimi.it}}

%%% in case of several authors:
%\smallskip
%\centerline{Author 2}
%\centerline{\it Institution Name, Country}
%\centerline{e-mail {\tt Author 2}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\par
Introduced in the late fifties of the 20th century,
Chang's MV-algebras are the Lindenbaum algebras of the infinite-valued
 calculus of \L ukasiewicz.
 Boolean algebras are idempotent MV-algebras.
 As explained in the monograph
 jointly authored with R. Cignoli and I. M. L. D'Ottaviano,
``Algebraic Foundations of Many-valued reasoning'',
 Kluwer Academic Publishers, 2000, MV-algebras have many
interesting relations with such diverse mathematical structures
as the AF C*-algebras of operators in infinite spin systems,
and the fans of toric geometry. Further,
MV-algebras are equivalent, as a category, to
lattice-ordered abelian groups with strong unit.
While the archimedean property of
strong units is not even definable in first-order logic,
MV-algebras form a variety.
  By definition, locally
finite MV-algebras  are those MV algebras arising as direct limits of
finite MV-algebras. As shown in recent joint work with  R. Cignoli and
E. Dubuc, locally finite MV-algebras   provide an
interesting dual counterpart
of multisets, extending Stone's duality.

\newpage

\centerline{\large \bf Closure Operations and Complete Embeddings of 
Residuated}

\centerline{\large \bf Lattices, and Completeness Theorems of
Substructural Logics}

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\centerline{Hiroakira Ono}
\centerline{\it Japan Advanced Institute of Science and Technology, Japan}
\centerline{e-mail {\tt ono@jaist.ac.jp}}

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%   Text of the abstract
We will show a theorem on the existence of complete 
embeddings of partially ordered monoids into complete residuated
lattices. Our theorem covers many of related results
on embeddings, and from this, various completeness theorems 
for substructural logics 
e.g. {\it Kripke completeness} by Ono-Komori, {\it algebraic completeness
of predicate logics} by Ono, {\it finite embeddability property} by
Blok-van Alten, and {\it standard completeness} by Montagna-Ono, follow.

\medskip

Here, we consider only {\it commutative, integral} residuated lattices 
(~or, simply {\it residuated lattices}~). A residuated lattice with
the monoid operation $\ast$ is {\it complete} if it is complete as a 
lattice and satisfies $(\bigcup x_i) \ast y = \bigcup (x_i \ast y)$.
A map $C$ on a residuated lattice {\bf A} is a {\it closure
operation} if $C$ satisfies: (1) \ $x \leq Cx$, (2) \ $x \leq y$ implies
$Cx \leq Cy$, (3) \ $CCx \leq Cx$, and (4) \ $Cx \ast Cy \leq 
C(x \ast y)$. Then the set $C({\bf A})$ of all $C$-closed elements of
${\bf A}$ forms a residuated lattice, and moreover if {\bf A} is complete
then so is $C({\bf A})$. 
Let {\bf M} be a {\em partially ordered commutative integral monoid} 
(~or, simply a {\em po-monoid}~). Then the power set $\wp({\bf M})$
of $M$ forms a complete residuated lattice where the monoid operation 
$\ast$ is defined by $X \ast Y = \{ a \cdot b \in M : a \in X$ and 
$b \in Y \}$. So, for any closure operation $C$ on $\wp({\bf M})$,
$C(\wp({\bf M}))$ is a complete residuated lattice. Conversely, any
complete residuated lattice can be represented as $C(\wp({\bf M}))$
for some po-monoid {\bf M} and some closure operation $C$ on $\wp({\bf M})$.

For a given po-monoid {\bf M} and a closure operation $C$ on $\wp({\bf M})$,
define a subset $U$ of $M$ by $U = \{ b \in M : (b]$ is $C$-closed$\}$, 
where $(b] = \{ x \in M : x \leq b \}$. We assume moreover that 
for any $C$-closed subset $X$ (1) $X$ is downward closed, and (2)
$\cup_i a_i \in X$ whenever $\{ a_i : i \in I \} \subseteq X \cap U$ 
and $\cup_i a_i \in U$. Then, the following is our main theorem: 

{\it The map $h: U \rightarrow C(\wp({\bf M}))$ defined by $h(b) = (b]$ is a 
complete embedding, i.e. an order isomorphism which 
preserves all exisiting products and residuals in $U$, and also all 
existing (infinite) joins and meets in $U$.}

\medskip

As mentioned above, our result has many logical consequences on
completeness of substructural logics. For instance, take any
residuated lattice {\bf A} and consider the Dedekind-McNeille
completion. Then our theorem implies the existence of a complete
embedding of {\bf A} into its Dedekind-McNeille completion. This
implies completeness theorem for {\it basic} substructural 
logics with respect to {\it phase structures}, and for substructural
predicate logics with respect to algebraic semantics.

\newpage
\centerline{\large \bf Sugihara Algebras and}

\centerline{\large \bf Consequence Fragments of R-Mingle}

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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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\par
%   Text of the abstract

The variety SA of Sugihara algebras and its various classes  of subreducts with residuation are the equivalent semantics  for the logic R-Mingle (briefly RM) and its respective fragments with  implication - considered as full consequence relations (rather than as sets of theorems).  For philosophical reasons these `consequence fragments' were not studied explicitly by relevance logicians, but they are completely susceptible to algebraic analysis.  

The basic algebraic properties of SA's subreduct classes will be established and applied, using methods of abstract algebraic logic, to give some new results for RM's consequence fragments.  For example, all are finitely axiomatized and have (at worst) a 2-term local deduction theorem.  In terms of their most interesting metalogical (or  algebraic) properties, they turn out to be naturally partitioned into two classes, according to whether or not disjunction is definable.  The fragments without disjunction are closely related to a congruence meet semidistributive variety generated by a 3-element algebra first studied by Sobocinski (1952), whose properties will also be discussed.  

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\newpage
\centerline{\large \bf Algebrizing the Algebrizable}

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\centerline{Aldo Ursini}
\centerline{\it University of Siena, Italy}
\centerline{e-mail ursini@unisi.it}


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\par
A survey of some recent and ongoing  researches in Siena on Algebraic
Logic; part A:  traditional algebraic logic; part
B:  abstract algebraic logic.

Part A.
{\it (Strong many--valued logics.)}
 The main example
 of a  many--valued
discriminator variety
is  MV algebras with  product and  residuations.
Results: a complete axiomatization;
 the variety is generated by the algebra on the unit interval $[0,1]$
with \L ukasiewicz conjunction, product and residuations; it is
 decidable;
 it is categorially equivalent to a class of $l-$rings which are
subdirect products of ordered fields with a suitable ideal J pointed out;
 the free algebras in the variety are completely described.
 (F.Montagna and collaborators)

{\it (Weak many--valued logics.) } BL algebras correspond to the logic of
continuous $t$-norms on $[0,1].$ Results: the variety is generated by finite ordinal sums of finite linearly
ordered Wajsberg hoops, and any BL algebra is an ordinal sum of
linearly ordered Wajsberg hoops; 
the algebras which generate the whole variety and the subdirectly
irreducible algebras are characterized.
For Esteva and Godo's logic MTL: the variety
is generated by residuated lattices on $[0,1]$ (with the usual  order; the
product is to be a left continuous $t--$norm);  the variety has the
finite embeddability property.(F.Montagna, P. Agliano and collaborators)

{\it (Positive classical logic.)} The algebras of positive classical logic
(i.e. without negation and the constant for Falsum) are
boolean ultrafilters (i.e. those lattices which are ultrafilters
 of a boolean
algebra). Equivalently, they are boolean rings (not necessarily with unit).
Results: several equational axiomatizations; decidability, but no quantifier elimination;
   topological
representation and duality.(A.Ursini)


Part B.{\it (Equational derivations.)} The common feeling that algebraic logic is not able to deal with
the main logical notion (i.e. that of a formal derivation) is wrong.
For most logics $L$ which have a constant $1$ for True and are algebrizable in a
class $V$ of algebras,
the following notion of a formula (i.e. term) $t$ being
{\it e-derivable from }
formulas $b_1,... b_n$ is suitable: that there is a term
 $p(x_1,\dots x_n,y_1,\dots,y_m)$ such that
 the identity $p(\vec x, 1,\dots,1)=1$ hold in $V,$ and that
 for some terms $u_1,\dots,u_n$, the identity $p(\vec u,\vec b)= t$
 hold in $V.$
One can deal with typical features of proof theory in
 equational logic: e.g. terms of typed Lambda calculus
 admit a reading as e-derivations in the equational theory of
 Heyting algebras. This generalizes in all cases an equational characterization
 of the class $V$ is given:  derivations in the logic
 $L$ correspond with e-derivations in the equational logic of $V.$ (A. Ursini)

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