On Lattices of Varieties of Restriction Semigroups

The left restriction semigroups have arisen in a number of contexts, one being as the abstract characterization of semigroups of partial maps, another as the ‘weakly left E-ample’ semigroups of the ‘York school’, and, more recently as a variety of unary semigroups defined by a set of simple identities. We initiate a study of the lattice of varieties of such semigroups and, in parallel, of their two-sided versions, the restriction semigroups. Although at the very bottom of the respective lattices the behaviour is akin to that of varieties of inverse semigroups, more interesting features are soon found in the minimal varieties that do not consist of semilattices of monoids, associated with certain ‘forbidden’ semigroups. There are two such in the one-sided case, three in the two-sided case. Also of interest in the one-sided case are the varieties consisting of unions of monoids, far indeed from any analogue for inverse semigroups. In a sequel, the author will show, in the two-sided case, that some rather surprising behavior is observed at the next ‘level’ of the lattice of varieties

On Semigroups with Lower Semimodular Lattice of Subsemigroups

The question of which semigroups have lower semimodular lattice of subsemigroups has been open since the early 1960s, when the corresponding question was answered for modularity and for upper semimodularity. We provide a characterization of such semigroups in the language of principal factors. Since it is easily seen (and has long been known) that semigroups for which Green\u27s relation J is trivial have this property, a description in such terms is natural. In the case of periodic semigroups—a case that turns out to include all eventually regular semigroups—the characterization becomes quite explicit and yields interesting consequences. In the general case, it remains an open question whether there exists a simple, but not completely simple, semigroup with this property. Any such semigroup must at least be idempotent-free and D-trivial

A note on the Howson property in inverse semigroups

An algebra has the Howson property if the intersection of any two finitely generated subalgebras is finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.Comment: 6 page

The Semigroups B\u3csub\u3e2\u3c/sub\u3e and B\u3csub\u3e0\u3c/sub\u3e are Inherently Nonfinitely Based, as Restriction Semigroups

The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B2 carries the natural structure of an inverse semigroup. Regarded as such, in the signature {⋅, -1}, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, forgetting the inverse operation x ↦ x-1 and retaining the induced operations x ↦ x+ = xx-1 and x ↦ x* = x-1x — it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based). The essence of the nonfinite behavior is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2. It is again inherently nonfinitely based, regarded in that fashion. It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based. Therefore for finite restriction semigroups, the existence of a finite basis is decidable modulo monoids . These results are consequences of — and discovered as a result of — an analysis of varieties of strict restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of completely r-semisimple restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation �. For example, explicit bases of identities are found for the varieties generated by B0 and B2

A Common Framework for Restriction Semigroups and Regular *-Semigroups

Left restriction semigroups have appeared at the convergence of several flows of research, including the theories of abstract semigroups, of partial mappings, of closure operations and even in logic. For instance, they model unary semigroups of partial mappings on a set, where the unary operation takes a map to the identity map on its domain. This perspective leads naturally to dual and two-sided versions of the restriction property. From a varietal perspective, these classes of semigroups–more generally, the corresponding classes of Ehresmann semigroups–derive from reducts of inverse semigroups, now taking a to a+=aa−1 (or, dually, to a∗=a−1a, or in the two-sided version, to both). In this paper the notion of restriction semigroup is generalized to P-restriction semigroup, derived instead from reducts of regular ∗-semigroups (semigroups with a regular involution). Similarly, [left, right] Ehresmann semigroups are generalized to [left, right] P-Ehresmann semigroups. The first main theorem is an abstract characterization of the posets P of projections of each type of such semigroup as ‘projection algebras’. The second main theorem, at least in the two-sided case, is that for every P-restriction semigroup S there is a P-separating representation into a regular ∗-semigroup, namely the ‘Munn’ semigroup on its projection algebra, consisting of the isomorphisms between the algebra’s principal ideals under a modified composition. This theorem specializes to known results for restriction semigroups and for regular ∗-semigroups. A consequence of this representation is that projection algebras also characterize the posets of projections of regular ∗-semigroups. By further characterizing the sets of projections ‘internally’, we connect our universal algebraic approach with the classical approach of the so-called ‘York school’. The representation theorem will be used in a sequel to show how the structure of the free members in some natural varieties of (P-)restriction semigroups may easily be deduced from the known structure of associated free inverse semigroups

Varieties of \u3cem\u3eP\u3c/em\u3e-Restriction Semigroups

The restriction semigroups, in both their one-sided and two-sided versions, have arisen in various fashions, meriting study for their own sake. From one historical perspective, as “weakly E-ample” semigroups, the definition revolves around a “designated set” of commuting idempotents, better thought of as projections. This class includes the inverse semigroups in a natural fashion. In a recent paper, the author introduced P-restriction semigroups in order to broaden the notion of “projection” (thereby encompassing the regular *-semigroups). That study is continued here from the varietal perspective introduced for restriction semigroups by V. Gould. The relationship between varieties of regular *-semigroups and varieties of P-restriction semigroups is studied. In particular, a tight relationship exists between varieties of orthodox *-semigroups and varieties of “orthodox” P-restriction semigroups, leading to concrete descriptions of the free orthodox P-restriction semigroups and related structures. Specializing further, new, elementary paths are found for descriptions of the free restriction semigroups, in both the two-sided and one-sided cases

Lower Semimodular Inverse Semigroups, II

The authors’ description of the inverse semigroups S for which the lattice ℒℱ(S) of full inverse subsemigroups is lower semimodular is used to describe those for which (a) the lattice ℒ(S) of all inverse subsemigroups or (b) the lattice �o(S) of convex inverse subsemigroups has that property. In each case, we show that this occurs if and only if the entire lattice is a subdirect product of ℒℱ(S) with ℒ(E S ), or �o(E S ), respectively, where E S is the semilattice of idempotents of S; a simple necessary and sufficient condition is found for each decomposition. For a semilattice E, ℒ(E) is in fact always lower semimodular, and �o(E) is lower semimodular if and only if E is a tree. The conjunction of these results leads to quite a divergence between the ultimate descriptions in the two cases, ℒ(S) and �o(S), with the latter being substantially richer

Entanglement entropy in top-down models

We explore holographic entanglement entropy in ten-dimensional supergravity solutions. It has been proposed that entanglement entropy can be computed in such top-down models using minimal surfaces which asymptotically wrap the compact part of the geometry. We show explicitly in a wide range of examples that the holographic entanglement entropy thus computed agrees with the entanglement entropy computed using the Ryu-Takayanagi formula from the lower-dimensional Einstein metric obtained from reduction over the compact space. Our examples include not only consistent truncations but also cases in which no consistent truncation exists and Kaluza-Klein holography is used to identify the lower-dimensional Einstein metric. We then give a general proof, based on the Lewkowycz-Maldacena approach, of the top-down entanglement entropy formula.Comment: 40 page

Entanglement entropy and differential entropy for massive flavors

In this paper we compute the holographic entanglement entropy for massive flavors in the D3-D7 system, for arbitrary mass and various entangling region geometries. We show that the universal terms in the entanglement entropy exactly match those computed in the dual theory using conformal perturbation theory. We derive holographically the universal terms in the entanglement entropy for a CFT perturbed by a relevant operator, up to second order in the coupling; our results are valid for any entangling region geometry. We present a new method for computing the entanglement entropy of any top-down brane probe system using Kaluza-Klein holography and illustrate our results with massive flavors at finite density. Finally we discuss the differential entropy for brane probe systems, emphasising that the differential entropy captures only the effective lower-dimensional Einstein metric rather than the ten-dimensional geometry.Comment: 54 pages, 8 figures; v2 references and comments adde