# Name for the concept where gluts are not (necessarily) a filter? (Or do gluts have to be a filter?)

In the following proof (which you'll excuse me, I hope, for giving only as slides/images), it is shown that (the relevance logic) R enriched with (A->B)\/(B->A) --enrichment which the authors call FR-- is still not as strong as RM, meaning that A->(A->A) (aka mingle) still fails in FR. An interesting aspect of this is that the gluts in R are "gappy" in the sense that both 1->1 and 3->3 so both 1 and 3 are possible/designated truth values, but in the model you see there (which is a model for FR, thus also for R, although it's a counter-model for RM) you have that 2-/->2, so 2 is not a truth/designated value. But you'll also note that 2 is in between 1 and 3, so basically the set or truth/designated values {1, 3} is not a filter (in the sense of lattice theory) in that model of R.

So, basically not only there are gluts and gaps in R... but the gluts themselves are "gappy"; in this case 2 is a "gap" in-between 1 and 3.

My question is basically/just: how is notion/occurrence usually called? "Gappy gluts"? Or am I just misreading that and 2 is also a possible designated value in that model of R (because 2->2=1; but in the proof 2->(2->2) = 2->1 = 0, which is really below any designated value)? I.e. do gluts have to be a filter?