Knowledge representation systems and skew nearlattices
by
Janis Cirulis
University of Latvia

The concept of a knowledge representation system (kr-system, for short) we deal with is a special case of that introduced in [2]. A frame is a pair (A, V), where A is a poset treated as a category, and V is a contravariant functor A --> Set respecting coproducts. Thus, V associates a set V_a with every a in A and a function V_b --> V_a with every pair (a, b) in V² such that a <= b. A kr-system is a quadruple (A, V, S, \Pi), where (A, V) is a frame, S is a set, and \Pi is a cone (\pi_a\colon S --> V_a) in Set from S to V. Elements of A and S may be thought of as attributes and states (a <= b means that a is a part of b), respectively, those of V_a as possible values of a, and \pi_a(s) as the value of a in the state s.

A right normal skew nearlattice (rnsn-lattice, for short) was defined in [1] as an algebra (L, \/ , \odot), where (A, \odot) is a right normal band possessing the upper bound property w.r.t. its natural ordering, and \/ is the corresponding partial join operation. A relative subalgebra of L is said to be commutative, if \odot is commutative on it. L is said to be locally commutative if every principal order ideal in it is commutative. An ideal of L is a downward closed subset closed also under existing joins (hence, a relative subalgebra of L).

A commutative skew nearlattice is known as a nearlattice [3]. We shall consider here only kr-systems with the attribute set a nearlattice.

Theorem. There is a one-to-one correspondence between frames and rnsn-lattices. Furthermore, every kr-system can be presented as a triple (W, L, \models), where W is a set, L is an rnsn-lattice, and \models is a relation on W ×L such that each set {x in L\colon w \models x} is a commutative ideal of L. Moreover, a kr-system is completely determined, up to isomorphism, by its triple.

References

[1] Cirulis, J., A class of skew nearlattices. Colloq. on Semigroups (Szeged, July 17-21, 2000), Abstracts, http//at.yorku.ca/c/a/e/c/18.htm .

[2] Cirulis, J., Are there essentially incomplete knowledge representation systems? Lect. Notes Comput. Sci 2138 (2001), 94-105.

[3] Cornish, W.H., 3-permutability and quasi-commutative BCK-algebras. Math. Japon. 25 (1980), 477-496.

Date received: January 28, 2003