L-effect algebras are introduced as a class of L-algebras which specialize to all known generalizations of effect algebras with a $$\wedge $$ -semilattice structure. Moreover, L-effect algebras X arise in connection with quantum sets and Frobenius algebras. The translates of X in the self-similar closure S(X) form a covering, and the structure of X is shown to be equivalent to the compatibility of overlapping translates. A second characterization represents an L-effect algebra in the spirit of closed categories. As an application, it is proved that every lattice effect algebra is an interval in a right $$\ell $$ -group, the structure group of the corresponding L-algebra. A block theory for generalized lattice effect algebras, and the existence of a generalized OML as the subalgebra of sharp elements are derived from this description.

中文翻译：

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L 效应代数

L-效应代数作为一类 L-代数被引入，它专门用于具有 $$\wedge $$ -半格结构的效应代数的所有已知推广。此外，L 效应代数 X 与量子集和 Frobenius 代数有关。X 在自相似闭包 S(X) 中的平移形成一个覆盖，并且 X 的结构被证明等效于重叠平移的兼容性。第二个特征代表了封闭范畴精神下的 L 效应代数。作为应用，证明了每一个格效应代数都是右$$\ell $$ -group 中的一个区间，即对应L-代数的结构群。广义晶格效应代数的块理论，以及广义 OML 作为尖锐元素的子代数的存在是从这个描述中推导出来的。