A Class of Skew Nearlattices
by
Jānis Cīrulis
University of Latvia

A nearlattice has been defined to be a meet semilattice possessing the upper bound property: every pair of elements having a upper bound has the least upper bound (see, e.g., [2]).

Now suppose that (A, \odot) is a right normal band and that < is the canonic ordering of A: x < y iff x \odot y = x. Let \/ stand for the partial join operation on A, and let x \flat y mean that x \/ y exists. We call the partial algebra (A, \/ , \odot) a right normal skew nearlattice (rnsn-lattice, for short) if (1) the poset (A, < ) has the upper bound property and (2) x \flat y whenever x \odot y = y \odot x. See [1] for general information on partial algebras, in particular, for the notion of an ECE-variety.

Proposition. The class of rnsn-lattices is an ECE-variety.

An rnsn-lattice is said to be distributive if the following holds:

if y \flat z, then (x \odot y) \flat (x \odot z) and x \odot (y \/ z) = (x \odot y) \/ (x \odot z).

Example. Consider the set PF(K, L) of partial functions from K to L. Given two functions j, \psi in PF(K, L), let j \/ \psi: = j \cup \psi provided that the union belongs to PF(K, L). Furthermore, let j\odot \psi = \psi| (\dom j \cap \dom \psi) (i.e., the restiction of \psi to the common part of domains of \phi and \psi). Then (PF(K, L), \/ , \odot) is a distributive rnsn-lattice.

We now define a rnsn-lattice of partial fumctions to be a closed subalgebra af an algebra of kind PF(K, L). The following theorem is an extension of a similar representation theorem for right normal bands (restrictive semigrops) in [3].

Theorem. An algebra (A, \/ , \odot) is a distributive rnsn-lattice if and only if it is isomorphic to a rnsn-lattice of partial functions.

References

[1] Burmeister,   P., Partial algebras - an introductory survey. In: I.G. Rosenberg, G. Sabidussi (eds.), Algebras and Orders. Kluwer Acad. Publ, 1993, 1-70.

[2] Noor,  A.S.A., Cornish,  W.H., Multipliers on a nearlattice. Comment. Univ. Carol. 27 (1986), 815-827.

[3] Vagner,   V.V., Restrictive semigroups. Izv. Vyssh. Uchebn. Zav. Mat. 1962, no. 6(31), 19-27 (Russian).

Date received: May 5, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-18.