Z-continuous posets.

*(English)*Zbl 0614.06007In [Lect. Notes Comput. Sci. 53, 192-212 (1977; Zbl 0372.06002)] J. B. Wright, E. G. Wagner and J. W. Thatcher suggested a way to generalize Dana Scott’s continuous lattices. There the authors introduced the concept of a Z-subset system.

This paper is an attempt to develop a theory of Z-continuous posets. Section 2 is mainly devoted to finding some natural classes of Z- continuous mappings under which the images of Z-continuous posets are Z- continuous. In this section the main result is theorem 2.15 which states that the image of a Z-continuous poset under a Z-continuous projection operator is Z-continuous. From this, as a corollary, the author gets the main result of H. J. Bandelt and M. ErnĂ© [J. Pure Appl. Algebra 30, 219-226 (1983; Zbl 0523.06001)]. In Section 3 he defines a basis for a Z-continuous poset and generalizes some results about the bases of continuous lattices. In this section he also studies the Z- algebraic posets. In the last section he considers the set of all Z- continuous extensions of an arbitrary poset B and defines a quasi-order on this set finally considering the poset arising from this quasi-ordered set in the usual way. He proves certain local and global results about this poset and studies the algebraic extensions of an arbitrary poset.

This paper is an attempt to develop a theory of Z-continuous posets. Section 2 is mainly devoted to finding some natural classes of Z- continuous mappings under which the images of Z-continuous posets are Z- continuous. In this section the main result is theorem 2.15 which states that the image of a Z-continuous poset under a Z-continuous projection operator is Z-continuous. From this, as a corollary, the author gets the main result of H. J. Bandelt and M. ErnĂ© [J. Pure Appl. Algebra 30, 219-226 (1983; Zbl 0523.06001)]. In Section 3 he defines a basis for a Z-continuous poset and generalizes some results about the bases of continuous lattices. In this section he also studies the Z- algebraic posets. In the last section he considers the set of all Z- continuous extensions of an arbitrary poset B and defines a quasi-order on this set finally considering the poset arising from this quasi-ordered set in the usual way. He proves certain local and global results about this poset and studies the algebraic extensions of an arbitrary poset.

Reviewer: R.A.Alo