Abstract
We introduce the following variant of an order and semigroup of quotients in inverse semigroups. A subsemigroup S of an inverse semigroup Q is a new left order in Q if for every q ∈ Q there exist a, b, ∈ S such that q = a-1b. Here a-1 is the unique inverse of a in Q. A new right order is defined dually, and a new order is the conjunction of the two. This concept produces more (left, right) orders in an inverse semigroup than those studied heretofore.
A primitive inverse semigroup is a nontrivial inverse semigroup with zero in which all nonzero idempotents are primitive. It can best be characterized as an orthogonal sum of Brandt semigroups.
Our main result consists of necessary and sufficient conditions on a semigroup S to be a new left order in a Brandt semigroup. They are five in number and of relatively concrete form. This result (with a long proof) is used to give two characterizations of new orders in Brandt semigroups, and eventually to perform a similar analysis for the same kinds of orders in primitive inverse semigroups. A uniqueness result concludes the work.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.Fountain Adequatesemigroups Proc. Edinburgh Math. Soc. 22 (1979), 113–12
J. Fountain, Abundant semigroups, Proc. London Math. Soc., 44 (1982), 103–129.
J. Fountain and M. Petrich, Brandt semigroups of quotients, Math. Proc. Cambridge Phil. Soc., 98 (1985), 413–426.
J. Fountain and M. Petrich, Completely 0-simple semigroups of quotients, J. Algebra, 101 (1986), 365–402.
J. Fountain and M. Petrich, Completely 0-simple semigroups of quotients III, Math. Proc. Cambridge Phil. Soc., 105 (1989), 263–275.
N. Ghroda and V. Gould, Semigroups of inverse quotients, Period. Math. Hungar., 65 (2012), 45–73.
V. Gould, Clifford semigroups of left quotients, Glasgow Math. J., 28 (1986), 181–191.
G. Lallement et M. Petrich, Décompositions I-matricielles d’un demi-groupe, J. Math. Pures Appl., 45 (1966), 67–117.
M. Petrich, Inverse semigroups, Wiley, New York, 1984.
Author information
Authors and Affiliations
Additional information
Communicated by M. B. Szendrei
The first author was partially supported by DGI of Spain, Project: MTM2011-22554. The authors are much indebted to Maria B. Szendrei and the referees, whose useful observations greatly improved our exposition
Rights and permissions
About this article
Cite this article
Cegarra, A.M., Ghroda, N. & Petrich, M. New orders in primitive inverse semigroup. ActaSci.Math. 81, 111–131 (2015). https://doi.org/10.14232/actasm-013-040-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.14232/actasm-013-040-7
Key words and phrases
- Brandt semigroup
- primitive inverse semigroup
- new (left) order
- semigroup of (left) quotients
- categorical at zero
- 0-cancellative