Page 98 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 98
NCOMMUTATIVE STRUCTURES WITHIN ORDER STRUCTURES, SEMIGROUPS
AND UNIVERSAL ALGEBRA (MS-67)
On modular skew lattices and their coset structure
Joao Pita Costa, joao.pitacosta@ijs.si
Institute Jozef Stefan, Slovenia
Modular lattices are of great importance in many branches of mathematics, from Algebra to
Topology. There are many well-known examples of lattices that are modular but not distributive:
the lattice of subspaces of a vector space, the modules over a ring, the normal subgroups of
a group, and many others. Modularity is a lattice property of somewhat topological flavour
where, in the finite case, maximal chains have the same size, and decomposition theorems as
the Krull-Schmidt-Remak for modules over a ring are derived from the modularity properties
of its subspace lattice. In this talk, we will discuss several approaches to the generalisation of
modularity to the non-commutative context of skew lattices, relate it to the known properties
of skew distributivity and skew cancellation, and derive topological-like properties from that
generalised concept. We will also discuss aspects of the coset structure of skew lattices of this
nature, and derive some of their combinatorial properties.
Noncommutativity in an algebraic theory of clones
Antonino Salibra, salibra@unive.it
Università Ca’Foscari Venezia, Italy
Coauthor: Antonio Bucciarelli
We introduce the notion of clone algebra, intended to found a one-sorted, purely algebraic
theory of clones. Clone algebras are defined by true identities and thus form a variety in the
sense of universal algebra. The most natural clone algebras, the ones the axioms are intended
to characterise, are algebras of functions, called functional clone algebras. The universe of a
functional clone algebra, called omega-clone, is a set of infinitary operations containing the
projections and closed under finitary compositions. We show that there exists a bijective cor-
respondence between clones (of finitary operations) and a suitable subclass of functional clone
algebras, called block algebras. Given a clone, the corresponding block algebra is obtained
by extending the operations of the clone by countably many dummy arguments. One of the
main results is the general representation theorem, where it is shown that every clone algebra
is isomorphic to a functional clone algebra. In another result we prove that the variety of clone
algebras is generated by the class of block algebras. This implies that every omega-clone is
algebraically generated by a suitable family of clones by using direct products, subalgebras and
homomorphic images. We present three applications. In the first one, we use clone algebras
to answer a classical question about the lattices of equational theories. The second application
is to the study of the category VAR of all varieties. We introduce the category CA of all clone
algebras (of arbitrary similarity type) with pure homomorphisms as arrows. We show that the
category VAR is categorically isomorphic to a full subcategory of CA. We use this result to
provide a generalisation of a classical theorem on independent varieties. In the third application
we show how skew Boolean algebras are related to clone algebras.
96
AND UNIVERSAL ALGEBRA (MS-67)
On modular skew lattices and their coset structure
Joao Pita Costa, joao.pitacosta@ijs.si
Institute Jozef Stefan, Slovenia
Modular lattices are of great importance in many branches of mathematics, from Algebra to
Topology. There are many well-known examples of lattices that are modular but not distributive:
the lattice of subspaces of a vector space, the modules over a ring, the normal subgroups of
a group, and many others. Modularity is a lattice property of somewhat topological flavour
where, in the finite case, maximal chains have the same size, and decomposition theorems as
the Krull-Schmidt-Remak for modules over a ring are derived from the modularity properties
of its subspace lattice. In this talk, we will discuss several approaches to the generalisation of
modularity to the non-commutative context of skew lattices, relate it to the known properties
of skew distributivity and skew cancellation, and derive topological-like properties from that
generalised concept. We will also discuss aspects of the coset structure of skew lattices of this
nature, and derive some of their combinatorial properties.
Noncommutativity in an algebraic theory of clones
Antonino Salibra, salibra@unive.it
Università Ca’Foscari Venezia, Italy
Coauthor: Antonio Bucciarelli
We introduce the notion of clone algebra, intended to found a one-sorted, purely algebraic
theory of clones. Clone algebras are defined by true identities and thus form a variety in the
sense of universal algebra. The most natural clone algebras, the ones the axioms are intended
to characterise, are algebras of functions, called functional clone algebras. The universe of a
functional clone algebra, called omega-clone, is a set of infinitary operations containing the
projections and closed under finitary compositions. We show that there exists a bijective cor-
respondence between clones (of finitary operations) and a suitable subclass of functional clone
algebras, called block algebras. Given a clone, the corresponding block algebra is obtained
by extending the operations of the clone by countably many dummy arguments. One of the
main results is the general representation theorem, where it is shown that every clone algebra
is isomorphic to a functional clone algebra. In another result we prove that the variety of clone
algebras is generated by the class of block algebras. This implies that every omega-clone is
algebraically generated by a suitable family of clones by using direct products, subalgebras and
homomorphic images. We present three applications. In the first one, we use clone algebras
to answer a classical question about the lattices of equational theories. The second application
is to the study of the category VAR of all varieties. We introduce the category CA of all clone
algebras (of arbitrary similarity type) with pure homomorphisms as arrows. We show that the
category VAR is categorically isomorphic to a full subcategory of CA. We use this result to
provide a generalisation of a classical theorem on independent varieties. In the third application
we show how skew Boolean algebras are related to clone algebras.
96
