997 research outputs found
Eilenberg Theorems for Free
Eilenberg-type correspondences, relating varieties of languages (e.g. of
finite words, infinite words, or trees) to pseudovarieties of finite algebras,
form the backbone of algebraic language theory. Numerous such correspondences
are known in the literature. We demonstrate that they all arise from the same
recipe: one models languages and the algebras recognizing them by monads on an
algebraic category, and applies a Stone-type duality. Our main contribution is
a variety theorem that covers e.g. Wilke's and Pin's work on
-languages, the variety theorem for cost functions of Daviaud,
Kuperberg, and Pin, and unifies the two previous categorical approaches of
Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new
results, including an extension of the local variety theorem of Gehrke,
Grigorieff, and Pin from finite to infinite words
Equidivisible pseudovarieties of semigroups
We give a complete characterization of pseudovarieties of semigroups whose
finitely generated relatively free profinite semigroups are equidivisible.
Besides the pseudovarieties of completely simple semigroups, they are precisely
the pseudovarieties that are closed under Mal'cev product on the left by the
pseudovariety of locally trivial semigroups. A further characterization which
turns out to be instrumental is as the non-completely simple pseudovarieties
that are closed under two-sided Karnofsky-Rhodes expansion
Varieties of Restriction Semigroups and Varieties of Categories
The variety of restriction semigroups may be most simply described as that generated from inverse semigroups (S, ·, −1) by forgetting the inverse operation and retaining the two operations x+ = xx−1 and x* = x−1x. The subvariety B of strictrestriction semigroups is that generated by the Brandt semigroups. At the top of its lattice of subvarieties are the two intervals [B2, B2M = B] and [B0, B0M]. Here, B2and B0 are, respectively, generated by the five-element Brandt semigroup and that obtained by removing one of its nonidempotents. The other two varieties are their joins with the variety of all monoids. It is shown here that the interval [B2, B] is isomorphic to the lattice of varieties of categories, as introduced by Tilson in a seminal paper on this topic. Important concepts, such as the local and global varieties associated with monoids, are readily identified under this isomorphism. Two of Tilson\u27s major theorems have natural interpretations and application to the interval [B2, B] and, with modification, to the interval [B0, B0M] that lies below it. Further exploration may lead to applications in the reverse direction
Weakly free regular semigroups I. The general case
We prove the existence of a regular semigroup F(X) weakly generated by X such
that all other regular semigroups weakly generated by X are homomorphic images
of F(X). The semigroup F(X) is introduced by a presentation and the word
problem for that presentation is solved.Comment: 23 pages; 4 figure
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
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