<ahref="javascript:toggleDiv('T34')">The class of representable semilattice-ordered monoids is not a variety</a>
<divstyle="display:none",id="T34"><br/><br/>We show a necessary and a sufficient condition for a quasi-variety to be a variety. Using this, we show that the quasi-variety of representable semilattice-ordered monoids is not a variety.</div>
<ahref="javascript:toggleDiv('T38')">Relational Sums and Splittings in Categories of L-fuzzy Relations</a>
<divstyle="display:none",id="T38"><br/><br/>Dedekind categories and similar structures provide a suitable framework to reason about binary relations in an abstract setting. Arrow categories extend this theory by certain operations and axioms so that additional aspects of L-fuzzy relations become expressible. In particular, arrow categories allow to identify crisp relations among all relations. On the other hand, the new operations and axioms in arrow categories force the category to be uniform, i.e., to be within a particular subclass of Dedekind categories. As an extension, arrow categories inherit constructions from Dedekind categories such as the definition of relational sums and splittings. However, these constructions are usually modified in arrow categories by requiring that certain relations are additionally crisp. This additional crispness requirement and the fact that the category is uniform raises a general question about these constructions in arrow categories. When can we guarantee the existence of the construction with and without the additional requirement of crispness in the given arrow category or or an extension thereof? This paper provides a complete answer to this complex question for the two constructions mentioned.</div>
<ahref="javascript:toggleDiv('T39')">FuReM - A System for Visualization and Manipulation of L-Fuzzy Relations (short talk)</a>
<divstyle="display:none",id="T39"><br/><br/>In this presentation we will introduce the FuReM (Fuzzy Relation Manipulator) sys- tem. This system allows to visualize and manipulate so-called L-fuzzy relations similar to the RelView system.</div>
<divstyle="display:none",id="T40"><br/><br/>Membership in the Class of Representable Tarskian Relation Algebra in undecid- able. However, dropping some operations to obtain a reduct language may introduce some more favourable properties. The Finite Representation Property (FRP) is a great example of good algebraic behaviour. A signature is said to have the FRP if every rep- resentable finite structure in that signature can be represented over a finite base set. Some positive implications of the FRP for a signature include the decidability of mem- bership in the representation class for finite structures, as well as the decidability of membership in equational theory generated by the representation class. Although the property fails for the language of Relation Algebras, it remains an open question for many of its reduct languages. We examine the conjecture that a reduct language of Re- lation Algebra has the FRP if and only if it does not include negation and composition, nor meet and composition. Here we prove that any signature containing composition and negation fails to have the FRP. This preliminary result, together with FRP failing for any signature containing composition and meet [Neu16], proves the right to left implication of the conjecture. For the left to right implication, the FRP is known to hold for composition-free signatures as well as a handful of those containing composi- tion. Furthermore, we show here that in any reduct signature with composition but neither meet nor negation a [finite] representable structure has a [finite] representation if and only if it embeds into a [finite] Relation Algebra (not necessarily representable). This suggests that well known counterexamples to FRP in other signatures, like the Point Algebra will have a finite base representation in signatures conjectured to have the FRP. Finally, we look at signatures, like that of join-lattice semigroups, where embedding a finite representable structure into a finite relation algebra appears to be especially difficult and look at examples of some of the more interesting known such embeddings.
<divstyle="display:none",id="T41"><br/><br/>In this short notice
we give some ideas how to compute isolated sublattices which can be used to derive a recursive algorithm for the computation of the number of closure operators on a finite lattice. We give an asymptotically optimal algorithm for deciding the existence and - in the case of existence - the computation of useful nontrivial isolated summit sublattices. The general case (i.e., an optimal algorithm for the computation of general nontrivial useful isolated sublattices) remains unsolved, however, we try to give some ideas and hints for future research.</div>
<divstyle="display:none",id="T42"><br/><br/>In weighted automata theory, many classical results on formal languages have been ex- tended into a quantitative setting. Here, we investigate weighted context-free languages of infinite words, a generalization of ω-context-free languages (Cohen, Gold 1977) and an extension of weighted context-free languages of finite words (Chomsky, Schützenberger 1963). As in the theory of formal grammars, these weighted context-free languages, or ω-algebraic series, can be represented as solutions of (mixed) ω-algebraic systems of equations and by weighted ω-pushdown automata.
In our first main result, we show that (mixed) ω-algebraic systems can be transformed into Greibach normal form. We use the Greibach normal form in our second main result to prove that simple ω-reset pushdown automata recognize all ω-algebraic series. Simple ω-reset automata do not use ε-transitions and can change the stack only by at most one symbol. These results generalize fundamental properties of context-free languages to weighted context-free languages.</div>
@@ -1047,6 +1053,7 @@ This work is part of an ongoing program to develop a theory of regular concurren
This talk is based on arxiv:2103.07557 which has recently been accepted for publication in MSCS. In the talk I will put this work in the context above.</div>
