Varieties of Structurally Trivial Semigroups II |
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Authors: | Kopamu |
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Institution: | (1) Department of Mathematics & Computer Science PNG University of Technology Private Mail Bag Lae, Papua New Guinea samkopamu@hotmail.com, PG |
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Abstract: | Abstract. Melnik 5] determined completely the lattice of all 2-nilpotent extensions of rectangular band varieties; and Koselev 4]
determined a distributive sublattice formed by certain varieties of n-nilpotent extensions of left zero bands. In 2] the
author described the skeleton of the lattice of all 3-nilpotent extensions of rectangular bands. We generalize these results
by proving that a certain family of semigroup varieties which includes all the varieties mentioned above, and referred to
here as planar varieties, consisting of certain n-nilpotent extensions of rectangular bands forms a distributive sublattice
that looks somewhat like an inverted pyramid. Our proof makes use of a countably infinite family of injective endomorphisms
on the lattice of all semigroup varieties that was introduced by the author in 1]. Although we do not determine completely
the lattice of all n-nilpotent extensions of rectangular band varieties, our result unifies certain previously known results
and provides a framework for further research. |
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