Classes of unars that are close to flat ones

This paper is devoted to the description of unars satisfying various conditions that are close to flatness. I.A.Sakharov showed that projective unars coincide with free ones and are a coproduct of rays. Previously, the author obtained a complete description of flat unars. This paper continues this line of research and provides a complete description of unars that are close to flat ones, namely: pullback flat, equalizer flat, weakly flat, principally weakly flat, torsion free, unars with conditions (P) or (E), faithful, strongly faithful and regular ones. It is proved that pullback flat and equalizer flat unars coincide and are a coproduct of lines and rays. Unars satisfying condition (P), flat, weakly flat, principally weakly flat and torsion free ones coincide and are a coproduct of lines, rays and cycles. Faithful, strongly faithful, regular unars and unars satisfying condition (E) are exactly unars that do not contain a cycle.