THE INTERNAL IDEAL LATTICE IN THE TOPOS OF M-SETS

Abstract

We believe that the study of the notions of universal algebra modelled in an
arbitarry topos rather than in the category of sets provides a deeper
understanding of the real features of the algebraic notions. [2], [3], [4], [S], [6],
[7], [13], [14] are some examples of this approach. The lattice Id(L) of ideals of
a lattice L (in the category of sets) is an important ingredient of the category of
lattices. In this paper, we construct the (internal) ideal lattice T(A) of a lattice
A in the topos of M-sets for a monoid M. The process of the construction of
Y(A) is so that it can also be done in any arbitrary topos whose ingredients are
known. Finally, we consider the lattice structure of T(A) for some special kind
of lattices A in the topos of M-sets and show, among other things, that if A is
an internally complete M-Boolean algebra then Y(A) is an M-Stone lattice