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 effect 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 $\odot$ is commutative. We get some interesting results for self-similar pseudo L-algebras: (1) The negative cone $G^-$ of an $\ell$-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 $(L, \rightarrow, \rightsquigarrow ,1)$, we can generate a free semigroup $(A, \ast)$ by the set $L\setminus \{1\}$. Furthermore, we let $S(L)=A\cup\{1\}$ and define a binary operation $\odot$ on $S(L)$. Then we extend the operations $\rightarrow$ and $\rightsquigarrow$ from $L$ to $S(L)$, and prove that $(S(L), \rightarrow, 1)$ and $(S(L), \rightsquigarrow, 1)$ are two cycloids, respectively. Furthermore, under some conditions, $(S(L), \rightarrow, \rightsquigarrow, 1)$ 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 $G(L)$.