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: (۱) The negative cone G^- of an \ell-group G can be seen as a self-similar pseudo L-algebra. (۲) 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 ,۱), we can generate a free semigroup (A, \ast) by the set L\setminus \{۱\}. Furthermore, we let S(L)=A\cup\{۱\} 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, ۱) and (S(L), \rightsquigarrow, ۱) are two cycloids, respectively. Furthermore, under some conditions, (S(L), \rightarrow, \rightsquigarrow, ۱) 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).

کلیدواژه ها:

Pseudo L-algebra ، self-similar closure of pseudo L-algebra ، structure group of pseudo L-algebra ، pseudo hoop ، ell-group