Pseudo L-algebras

We introduce generalized structures of L-algebras, called pseudo L-algebras, which are the multiplication reduct of pseudo hoops and are structures combining two L-algebras with one compatible order. We prove that every pseudo hoop gives rise to a pseudo L-algebra and every pseudo e ect algebra gives rise to a pseudo L-algebra. The self-similarity is the most important property of an L-algebra L, which guarantees to induce a multiplication on L. We introduce a notion of self-similar pseudo L-algebras and prove that a self-similar pseudo L-algebra becomes an L-algebra if and only if the multiplication ⊙ is commutative. We get some interesting results for self-similar pseudo L-algebras: (1) The negative cone G- of an ℓ-group G can be seen as a self-similar pseudo L-algebra. (2) Every self-similar pseudo L-algebra is a pseudo hoop. Next, we introduce the notion of self-similar closures of pseudo L-algebras and obtain a self-similar closure by a recursive method. Given a pseudo L-algebra (L,→, , 1), we can generate a free semigroup (A, ∗) by the set L\{1}. Furthermore, we let S(L) = A∪{1} and de ne a binary operation ⊙ on S(L). Then we extend the operations → and from L to S(L), and prove that (S(L),→, 1) and (S(L), , 1) are two cycloids, respectively. Furthermore, under some conditions, (S(L),→, , 1) becomes a self-similar pseudo L-algebra. Finally, we introduce the notion of the structure group of pseudo L-algebras, and give an interesting example to show how to extend a pseudo L-algebra L into the pseudo self-similar closure S(L), and furthermore, derive it s structure group G(L).