Residuation groupoids -- again (Q1024711)

The present paper arose from contributions published by the author decades ago culminating in his main embedding result: any \(\ell\)-group cone admits a canonical embedding into some brick, and any cone algebra (in particular, any brick) admits a canonical embedding into some \(\ell\)-group cone. In this paper, a new method of embedding an \(\ell\)-group cone into an associated quotient \(\ell\)-group is provided. Also, the former embedding procedure is simplified for the particular case of a brick. Furthemore, semiclans (i.e., special semilattices endowed with a partial multiplication which is left positive with respect to the induced order) are shown to be bricks (and conversely). As a consequence, an alternative embedding method for semiclans is given. Finally, commutative bricks (bounded BCK-algebras) are considered elaborating a short cut of the first embedding step given in the embedding procedure for semiclans.