Languages of higher-dimensional automata

We introduce languages of higher-dimensional automata (HDAs) and develop some of their properties. To this end, we define a new category of precubical sets, uniquely naturally isomorphic to the standard one, and introduce a notion of event consistency. HDAs are then finite, labeled, event-consistent precubical sets with distinguished subsets of initial and accepting cells. Their languages are sets of interval orders closed under subsumption; as a major technical step, we expose a bijection between interval orders and a subclass of HDAs. We show that any finite subsumption-closed set of interval orders is the language of an HDA, that languages of HDAs are closed under binary unions and parallel composition, and that bisimilarity implies language equivalence.

Fuzzy Alexandrov Topologies Associated to Fuzzy Interval Orders

We characterize the fuzzy T0 - Alexandrov topologies on a crisp set X, which are associated to fuzzy interval orders R on X. In this way, we generalize a well known result by Rabinovitch (1978), according to which a crisp partial order is a crisp interval order if and only if the family of all the strict upper sections of the partial order is nested.

Hereditary Semiorders and Enumeration of Semiorders by Dimension

In 2010, Bousquet-Mélou et al. defined sequences of nonnegative integers called ascent sequences and showed that the ascent sequences of length $n$ are in one-to-one correspondence with the interval orders, i.e., the posets not containing the poset $\mathbf{2}+\mathbf{2}$. Through the use of generating functions, this provided an answer to the longstanding open question of enumerating the (unlabeled) interval orders. A semiorder is an interval order having a representation in which all intervals have the same length. In terms of forbidden subposets, the semiorders exclude $\mathbf{2}+\mathbf{2}$ and $\mathbf{1}+\mathbf{3}$. The number of unlabeled semiorders on $n$ points has long been known to be the $n$th Catalan number. However, describing the ascent sequences that correspond to the semiorders under the bijection of Bousquet-Mélou et al. has proved difficult. In this paper, we discuss a major part of the difficulty in this area: the ascent sequence corresponding to a semiorder may have an initial subsequence that corresponds to an interval order that is not a semiorder. We define the hereditary semiorders to be those corresponding to an ascent sequence for which every initial subsequence also corresponds to a semiorder. We provide a structural result that characterizes the hereditary semiorders and use this characterization to determine the ordinary generating function for hereditary semiorders. We also use our characterization of hereditary semiorders and the characterization of semiorders of dimension $3$ given by Rabinovitch to provide a structural description of the semiorders of dimension at most $2$. From this description, we are able to determine the ordinary generating function for the semiorders of dimension at most $2$.