Abstract
For arbitrary monoidsAandB, in Cevik et al. (Hacet J Math Stat 2019:1-11, 2019), it has been recently defined an extended version of the general product under the name ofa higher version of Zappa products for monoids(orgeneralized general product)A(circle plus B) (delta)(sic)(psi) B-circle plus A and has been introduced an implicit presentation as well as some theories in terms of finite and infinite cases for this product. The goals of this paper are to present some algebraic structures such as regularity, inverse property, Green's relations over this new generalization, and to investigate some other properties and the product obtained by a left restriction semigroup and a semilattice.