7,849 research outputs found
The algebra of adjacency patterns: Rees matrix semigroups with reversion
We establish a surprisingly close relationship between universal Horn classes
of directed graphs and varieties generated by so-called adjacency semigroups
which are Rees matrix semigroups over the trivial group with the unary
operation of reversion. In particular, the lattice of subvarieties of the
variety generated by adjacency semigroups that are regular unary semigroups is
essentially the same as the lattice of universal Horn classes of reflexive
directed graphs. A number of examples follow, including a limit variety of
regular unary semigroups and finite unary semigroups with NP-hard variety
membership problems.Comment: 30 pages, 9 figure
Three-page encoding and complexity theory for spatial graphs
We construct a series of finitely presented semigroups. The centers of these
semigroups encode uniquely up to rigid ambient isotopy in 3-space all
non-oriented spatial graphs. This encoding is obtained by using three-page
embeddings of graphs into the product of the line with the cone on three
points. By exploiting three-page embeddings we introduce the notion of the
three-page complexity for spatial graphs. This complexity satisfies the
properties of finiteness and additivity under natural operations.Comment: 32 pages with 9 figures, submitted to J.Knot Theory and Ra
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