Obviously, a dense complete lattice has the following property: { α , α );( α , α ) < ( β , β ) }=( β , β );{ ( α , α );( β , β ) }=( β , β ).

Definition 8.24 Assume ( ( K , K ), )isasun-orderedset;mappingN;( K , K ) ( ( K , K ), )isasub-oderedset;mappingN:( K , K ) is order preserving mapping. When ( a , a )( b , b )(( a , a ),( b , b ) ( K , K ),thereisN( b , b )N( a , a ), N( N( a , a ) )=( a , a ),, and we call N as reverse mapping. If i reverse mapping N satisfies the law of unity, namely, N( N( a , a

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