Orders in normal bands of groups

On Orders in Normal Cryptogroups

“…Continuing in the line initiated in [2] and [3], paper [4] contains the case when C is the class of normal cryptogroups (i.e., normal bands of groups). This represents the next step after completely simple semigroups since normal cryptogroups are precisely strong semilattices of completely simple semigroups.…”

“…For example, if a semigroup S has a completely (0-)simple semigroup of quotients, it is unique up to an isomorphism. This property manifestly fails for orders in normal cryptogroups as noted in [4].…”

“…The phenomenon of non-uniqueness of the semigroups of quotients forces one of the following two ways out: either to strengthen the definition of an order or to gain some overview of all semigroups of quotients of a given semigroup. The first approach was adopted in [5] and [4], for example by the introduction of "stratified" and "distinguished" orders, respectively which aims at excluding the study of those orders which do not behave well under the requirement of uniqueness. The second approach turns our goals around.…”

“…Preliminaries are treated in Sect. 3 consisting mostly of relevant results from [2] and [4]. Section 4 contains a study of restrictions of Green's relations on a normal cryptogroup Q to an order S in Q, including the comparison of these for two normal cryptogroups of quotients of a fixed semigroup.…”

Orders and Straight Left Orders in Completely Regular Semigroups

“…If Q is completely regular, then H is a congruence on Q if and only if Q is a band of groups. Orders in bands of groups are the topic of [6]. In Section 4 we show that every order in a completely regular semigroup is straight as a left order (and dually as a right order).…”