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Keywords:
trellis; weakly-associative lattice; pseudo-ordered set; standard ideal; distributive ideal; congruence relation
trellis; weakly-associative lattice; pseudo-ordered set; standard ideal; distributive ideal; congruence relation
Summary:
The concept of standard ideals in lattices, introduced by G. Grätzer and E. T. Schmidt, plays a significant role in the development of lattice theory. It serves as a generalization of normal subgroups in groups and ideals in rings to lattices. Similarly, the notion of distributive ideals, introduced by O. Ore, is another important tool in lattice theory. Building on these ideas, we extend the concepts of standard ideals and distributive ideals to the broader context of trellises. A trellis is a pseudo-ordered set where every pair of elements has both the least upper bound and the greatest lower bound. Within this framework, many of the results of Grätzer and Schmidt are effectively generalized to trellises.
References:
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