A Generalized Class of Bisimple Ample Semigroups

It is clear that most results in ample semigroups are but analogues of inverse semigroups. Unlike bisimple inverse 𝝎-semigroups which 𝓗-classes contains regular elements as studied in [28] and later extended in [2] and [3] to a class of ample semigroups called ∗ - bisimple Ample 𝝎-semigroup and ∗ - simple Ample 𝝎-semigroup, there exists a class of ∗ - bisimple Ample 𝝎-semigroups in which certain 𝓗∗ -classes contains no regular elements. Close look at the internal structure of this class of Ample 𝝎-semigroups reveals that some of the 𝓗∗ -classes rather contains bisystems of cancellative monoids. However, the presence of these bisystems of cancellative monoids makes this class of semigroups different from the once studied in [28], [2], [3] and [22]. Thus, in this work, we study such a class of ∗ - bisimple Ample 𝝎-semigroups as an extension of the binary array of bisystems of cancellative monoids. The array of bisystems were closed and then certain rules are imposed to ensure the closure of multiplication of elements in the binary array of bisystems. Thus, we construct and study few ofits properties and then characterize them as a special extension of binary array of bisystems of sequence of cancellative monoids.