Boole代数与双格半群

Boole Algebra and Double Lattice-Orderd Semigroup

指出Boole代数类是双格半群类的真子类;有限Boole代数类是F 格半群类的真子类;当格群是Boole代数时,该格群一定是平凡的,同时给出一个双格半群(S, ,≤)是Boole代数的充要条件是:1 0∈S, x∈S,0≤x,0 x=x 0=x;2 x,y∈S,(x⊙y) x=x;3 x∈S, x′∈S,x⊙x′=x;4 x,y,z∈S,x y=x z,x⊙y=x⊙z x=y.

The class of Boole Algebra is a true subclass of the double lattice-ordered semigroup. The class of finite Boole Algebra is a true subclass of the class of F lattice-ordered semigroup. Let the lattice-ordered group is a Boole Algebra if and only if we have the following:1.0∈S,x∈S,0≤x, and 0x=x0=x(or 1∈S,x∈S,x≤1 and 1x=x1=1);2.x,y∈S,(x⊙y)x=x(or(xy)⊙x=x);3.x∈S,x⊙x′=0(or xx′=1),the mapping ′:S→S is an ordered-reversing involution on S;4.x,y,z∈S,if xy=xz,x⊙y=x⊙z,then y=z.