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 $q\in Q$, there exist $a,b\in S$ such that $q=a^{-1}b$. 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.

DOI: 10.14232/actasm-013-040-7

AMS Subject Classification (1991): 20M10, 20M18

Keyword(s): Brandt semigroup, primitive inverse semigroup, new (left) order, semigroup of (left) quotients, categorical at zero, $0$-cancellative

Received July 2, 2013, and in final form December 12, 2014. (Registered under 40/2013.)