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Full-Text Articles in Mathematics
Varieties Of Restriction Semigroups And Varieties Of Categories, Peter Jones
Varieties Of Restriction Semigroups And Varieties Of Categories, Peter Jones
Mathematics, Statistics and Computer Science Faculty Research and Publications
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 …
Almost Perfect Restriction Semigroups, Peter R. Jones
Almost Perfect Restriction Semigroups, Peter R. Jones
Mathematics, Statistics and Computer Science Faculty Research and Publications
We call a restriction semigroup almost perfect if it is proper and the least congruence that identifies all its projections is perfect. We show that any such semigroup is isomorphic to a ‘W -product’ W(T,Y)W(T,Y), where T is a monoid, Y is a semilattice and there is a homomorphism from T into the inverse semigroup TIYTIY of isomorphisms between ideals of Y. Conversely, all such W-products are almost perfect. Since we also show that every restriction semigroup has an easily computed cover of this type, the combination yields a ‘McAlister-type’ theorem for all restriction semigroups. …
Varieties Of P-Restriction Semigroups, Peter R. Jones
Varieties Of P-Restriction Semigroups, Peter R. Jones
Mathematics, Statistics and Computer Science Faculty Research and Publications
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 …
The Semigroups B2 And B0 Are Inherently Nonfinitely Based, As Restriction Semigroups, Peter R. Jones
The Semigroups B2 And B0 Are Inherently Nonfinitely Based, As Restriction Semigroups, Peter R. Jones
Mathematics, Statistics and Computer Science Faculty Research and Publications
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+ …
On Lattices Of Varieties Of Restriction Semigroups, Peter R. Jones
On Lattices Of Varieties Of Restriction Semigroups, Peter R. Jones
Mathematics, Statistics and Computer Science Faculty Research and Publications
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 …