[For the entire collection see Zbl 0621.00012.]
It is proved that a locally finite variety V of semigroups has the strong amalgamation property if and only if, for some n, one of the following two systems of identities \(\{x^{n+1}y=xy=xy^{n+1}\), \(xyzt=xzyt\}\), \(\{x^ 2yx=xyx^ 2\), \(xz=x(yx)^ nz=xz^{n+1}\}\) is satisfied in V. Such a V is either a subvariety of inflations of orthodox normal bands of abelian groups of finite exponent or a subvariety of inflations of completely simple semigroups over abelian groups of finite exponent. Also a set of generators of a locally finite variety of semigroups with strong amalgamation property is described.