Structural Analysis of the Union Operation in Fuzzy Semigraphs

We define a union operation for fuzzy semigraphs and prove that it is well defined when the underlying crisp union is a semigraph. Under this compatibility condition, common consecutive pairs and partial edges occur precisely within shared semigraph edges, while the consecutive-pair and induced memberships on nonshared edges remain unchanged. We obtain inclusion–exclusion formulas for order and size and an exact formula for the edge-membership sum. The latter separates the overlap correction from a nonnegative shared-edge min–max gap, which measures the difference between taking the minimum after pointwise maxima and taking the maximum of the original edge minima. We characterize when this gap vanishes. For two uniform edge fuzzy semigraphs of the same level, that level is preserved exactly when every shared edge has a consecutive pair attaining the minimum in both summands.