### Large Classes of Algebras

#### Michael Adams

A class **K** of algebras of similar type is a *quasivariety* provided ** K** = **ISPP _{u}**(

**K**), where

**I**(

**K**),

**S**(

**K**),

**P**(

**K**), and

**P**(

_{u}**K**) respectively denote the classes of all isomorphic images, subalgebras, direct products, and ultraproducts of algebras in

**K**. Every variety (that is, a class of algebras of similar type for which

**K**=

**HSP**(

**K**), where

**H**(

**K**) denotes the class of all homomorphic images of algebras in

**K**) is a quasivariety, but not every quasivariety is a variety.

Two notions of universal will be discussed:

A quasivariety **V** of algebras is * universal* (in the sense of Hedrlín and Pultr) if every
category of algebras of finite type (or, equivalently, the category
**G** of all connected directed graphs together with all
compatible mappings) is isomorphic to a full subcategory of **V**. If an embedding of **G** may be effected by a functor
**Φ**:**G**-> **V** which assigns a finite
algebra to each finite graph, then **V** is said to be * finite-to-finite universal*.

For a quasivariety **V**, let L(**V**) denote the lattice
(ordered by set inclusion) of all quasivarieties contained in
**V**. A quasivariety **V** is *Q-universal* (in the sense of Sapir) providing
that, for any quasivariety **W** of finite type, L(**W**) is a homomorphic image of a sublattice of L(**V**).

An historical background will be given for each notion, as well as the relationship between them. Recent results will also be presented.

### António A. R. Monteiro and the mathematical research in Portugal during the forties [twentieth century]

#### Elza Amaral

The first steps towards scientific research in Portugal were given during the middle of the thirties, in the twentieth century. During the decade 1936-1947, the Portuguese mathematical community encountered, for the first time a set of initiatives propitious to the scientific research: groups of study working in new Mathematical Centers, created by the *Instituto para a Alta Cultura*; the foundation of the *Portuguese Mathematical Society*; and the appearance of two mathematical journals: *Portugaliae Mathematica*, for the publication of original papers and *Gazeta da Matemática* devoted to the dissemination of mathematics movement and the renewal of methods and themes of study. The powerful activity of António A. R. Monteiro was one of the most important pieces of this great engine towards the development of research in Portugal.

### La matemática de António Monteiro

#### Roberto Cignoli

Se comentará la evolución de la investigación matemática de Monteiro desde su tesis doctoral en la Sorbona hasta sus influyentes trabajos en álgebra de la lógica, destacando su preocupación por los métodos finitísticos.

### The three-valued logic of quadratic form theory over real rings

#### Dickman

### History of the Representations of De Morgan Lattices

#### J. Michael Dunn

In 1957 Biaynicki-Birula and Rasiowa gave a representation for what they called "quasi-Boolean algebras." Quasi-Boolean algebras are bounded distributive lattices with an order-inverting unary operation of period two (think "negation"). In 1960 Monteiro introduced what he termed "De Morgan lattices" and in this and later papers Monteiro and his students and co-workers proved a number of significant results about De Morgan lattices. De Morgan lattices and quasi-Boolean algebras are essentially the same except that the first is required to be bounded.. I, and it seems most others, prefer the label "De Morgan lattice" to "quasi-Boolean algebra".

Throughout my career, starting with my dissertation (1966), I have given a number of representations of De Morgan lattices and related structures, and shown that these representations of De Morgan lattices are all effectively equivalent to the Biaynicki-Birula and Rasiowa representation. One of these in particular leads to the "Belnap-Dunn 4-valued logic". In this talk I will compare and contrast these various representations with respect to their intuitive interpretation, and end with somewhat speculative observations about how to extend them to capture further nuances of what I am calling information delegation. "One source tells me P, another source tells me not-P. What am I to believe? How am I to behave? What do I infer?" This has possible applications to the Semantic Web since we all know that contradictions abound on the Web. It also has potential applications to shared intelligence.

### On Monteiro's symmetric Heyting algebras and weak contractive fuzzy logics

#### Francesc Esteva and Lluís Godo

In the setting of logical systems associated to residuated lattices, Ono defined in [9] (see also [2]) the weak contractive systems corresponding to pseudo-complemented residuated lattices. Intuitionistic logic (corresponding to Heyting algebras) and infinitely-valued Gödel logic (corresponding to linear Heyting algebras) are remarkable examples of weak contractive systems. In these systems, the definable negation ¬φ as φ→0 is not involutive (it is in fact Gödel negation over linearly ordered algebras). In this setting, it makes sense to study the issue of adding an involutive negation to weak contractive systems. This is in fact what Monteiro did in his excellent monograph [8] for the case of Heyting algebras (the resulting structures were called Symmetric Heyting algebras), with special attention to the linear case. In the talk, after a general presentation of the t-norm based (fuzzy) logics as pre-linear extensions of Höhle's Monoidal logic [7] or Ono's FL_{ew}, we will study similar expansions for any weak contractive fuzzy logic, following the line of the paper [4] where the authors studied the logics obtained by adding an involutive negation to Godel and Product logics. This line has been further investigated in [1] and [6]. We prove that some of Monteiro's results are also valid in the general setting of pseudocomplemented residuated lattices and lead to new axiomatic presentations. Finally, we will also consider the issue of expansions with truth-constants of these logics with an additional involutive negation, following the pioneering work of Pavelka [10] and the more recent ones [5, 3, 11]. Standard and canonical completeness of these logics will be discussed as well.

**References**

[1] P. Cintula, E.P. Klement, R. Mesiar, and M. Navara. Fuzzy logics with an additional involutive negation. To appear in Proceedings of Linz Seminar 2005, 2007.

[2] N. Galatos, P. Jipsen, T. Kowalski and H. Ono Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Studies in Logic and the Foundations of Mathematics, Vol. 151, Elsevier, 2007.

[3] F. Esteva, J. Gispert, L. Godo, C. Noguera. Adding truth constants to logics of continuous t-norm: axiomatization and completeness results. Fuzzy Set and Systems 158, 597?618, 2007.

[4] F. Esteva, L. Godo, P. Hajek, M. Navara. Residuated fuzzy logics with an involutive negation. Archive for Mathematical Logic, 39(2):103?124, 2000.

[5] F. Esteva, L. Godo and C. Noguera. On rational Weak Nilpotent Minimum logics. Journal of Multiple-Valued Logic and Soft Computing, 12(1?2):9?32, 2006.

[6] T. Flaminio and E. Marchioni. T-norm based logics with an independent involutive negation. Fuzzy Sets and Systems, 157(4):3125?3144, 2006.

[7] U. Hohle. Commutative, residuated l-monoids. In: U. Hohle and E.P. Klement eds., Non-Classical Logics and Their Applications to Fuzzy Subsets, Kluwer Acad. Publ., Dordrecht, 1995, 53-106.

[8] A. Monteiro. Sur les alg`ebres de Heyting symetriques, Portugalia Mathematica 39, Fasc. 1-4 (1980) 1-237.

[9] H. Ono. Logic without contraction rule and residuated lattices I. Manuscript.

[10] J. Pavelka. On Fuzzy Logic I, II, III. Zeitschrift fur Math. Logik und Grundlagen der Math. 25 (1979) 45-52, 119-134, 447-464.

[11] P. Savicky, R. Cignoli, F. Esteva, L. Godo and C. Noguera. On product logic with truth constants. Journal of Logic and Computation, 16(2):205-225, 2006.

### Semi-Heyting Algebras: An Abstraction from Heyting algebras

#### H. P. Sankappanavar

In this talk I would like to present a new abstraction from Heyting algebras, which I call "Semi-Heyting algebras", as well as some related algebras. I was led to the discovery of these algebras in 1983-84 while writing [2]. Early results were presented in the AMS Abstracts in January, 1985 (see [3]).

It turns out that semi-Heyting algebras share with Heyting algebras some rather interesting properties; for example, they are distributive pseudocomplemented lattices and their congruences are determined by filters, to mention just a few. The variety of semi-Heyting algebras is rich for its "non-Heyting" examples. For instance, there are ten (non-somorphic) semi-Heyting algebras on a 3-element chain, of which, only one, of course, is a Heyting algebra. The other, nine, 3-element semi Heyting algebras might be of interest to logicians interested in many-valued logics, as, in some of them, "F" implies "T" is "F" (or something else).

Some results on the lattice of subvarieties will be mentioned. In particular, the equational bases for the 3-element semi-Heyting algebras will be presented. I also mention some expansions of semi-Heyting algebrs obtained by adding various "negation" operations. I conclude the talk by mentioning some open problems for future research.

**References**

[1] H.P. Sankappanavar, Semi-Heyting Algebras: An abstraction from Heyting algebras. Manuscript.

[2] H.P. Sankappanavar, Heyting algebras with dual pseudocomplementation, Pacific J. Math. 117 (1985), 405-415.

[3] H.P. Sankappanavar, Semi-Heyting Algebras, Amer. Math. Soc. Abstracts, January 1985, page 13.

### Some aspects of bounded BCK- algebras

#### Antoni Torrens

Las álgebras BCK acotadas (bounded BCK-algebras) fueros introducidas por K. Iseki en 1972 ([9]), con el fin de dar un marco algebraico para el cálculo implicacional de Meredith, de la Lógica combinatoria BCK, con una negación dada por la contradicción. Se definen a partir de las álgebras BCK (que solo tienen implicación), añadiendo una constante. De hecho, las álgebras BCK acotadas son la contrapartida de la Lógica BCK con una negación que cumple la ley de Duns Scoto, concretamente son su semántica algebraica equivalente en el sentido de [1].

Las álgebras BCK (BCK-algebras) fueron introducidas en [7, 8] y exhaustivamente estudiadas en las décadas de los 70 y 80, en la que ya aparecen artículos con amplios análisis de ellas, como por ejemplo [10] en 1972 y [4] en 1982. Un punto álgido de su estudio se da en el artículo [2], en el que se clasifican las variedades de BCK-álgebras, y que además contiene una excelente visión de la literatura de las BCK-álgebras existente hasta 1995.

A partir de entonces, las BCK-álgebras junto con los pocrims, han motivado en gran medida alguno de los estudios mas recientes sobre cuasivariedades.

La gran atención prestada a las BCK-álgebras contrasta con la poca atención que se han merecido las BCK-álgebras acotadas. Nuestro propósito es llamar la atención sobre estas álgebras presentando algunos resultados obtenidos recientemente por el autor conjuntamente con otros autores. Ver [3] [5] y [6], especialmente los contenidos en este último.

Esencialmente los resultados que se presentan, son sobre la representabilidad de las bounded BCK-algebras en productos subdirectos, productos directos y productos booleanos. En este último caso analizamos la representabilidad en función de si sus fibras son directamente indescomponibles, simples...etc.

**Referencias**

[1] Blok, W. J. and Pigozzi, D., *Algebraizable Logics*, Mem. Amer. Math. Soc., 77 (1989), N. 396, vii + 78 pp.

[2] W. Blok and J.G. Raftery. *On the quasivariety of BCK-algebras and its subvarieties*. Algebra Universalis, 33(1995), 68-90.

[3] R. Cignoli and A. Torrens, *Glivenko like theorems in natural expansions of BCK-logics*. Math. Log. Quart. 50(2004) 2, 111-125.

[4] W.H. Cornish, *On Iseki's BCK-algebras*, Lec. Not. Pure and Appl. Math. 74 (1982), 101-122.

[5] J. Gispert and A. Torrens, *Bounded BCK-algebras and their generated variety*. Mathematical Logic Quarterly 53 (2007), 206-213.

[6] J. Gispert and A. Torrens, *Boolean representations of Bounded BCK-algebras*. Manuscrito.

[7] Y. Imai and K.Iseki, *On axiom systems of Propositional calculi* , XIV Proc. Jap. Acad. , 42 (1966), 19-22

[8] K. Iseki, *An algebra related with a propositional calculus*. XIV Poc. Japan Acad. 42 (1966), 26-29.

[9] K. Iseki, *On bounded BCK-algebras*. Math. Sem. Notes Kobe University 3 (1976), 23-33, see also pages 34-36.

[10] K. Iseki and S. Tanaka, *An introduction to the theory of BCK-algebras*, Math. Japonica Acad. 23 (1978), 1-26.