*(English)*Zbl 0880.20051

Summary: Abel-Grassmann groupoids (or in some papers $LA$-semigroups) are considered in a few papers. In those papers some elementary properties are given. We shall try here to collect all results which are archieved in studying this class of groupoids. Most of these results are due to the authors and they are not published yet. There are also a few results of other mathematicians.

In the first section we present some general properties of Abel-Grassmann groupoids, $AG$-test and describe ideals in $AG$-groupoids. In the second section we deal with $AG$-bands. Natural partial order relations on $AG$-bands will be given, as well as decomposition of $AG$-bands. The third section is devoted to $A{G}^{*}$-groupoids. We give semilattice decomposition, natural partial order relation and structural theorem for this class of $AG$-groupoids. In the fourth section we describe $A{G}^{*}$-groupoids. Natural partial order, semilattice decomposition and congruences on $A{G}^{**}$-groupoids will be described.

##### MSC:

20N02 | Sets with a single binary operation (groupoids) |