Positive Sugihara monoids (Q995381)

Sugihara monoids are the distributive lattice-ordered commutative idempotent monoids that are residuated and have an involution operator; they form the equivalent algebraic semantics for the relevance logic \(\mathbf{RM}^t\). Positive Sugihara monoids are the residuated lattice-ordered monoids without involution that can be embedded into Sugihara monoids. Such algebras form a variety \(\mathsf {PSM}\), which is the equivalent algebraic semantics of the full negation-free fragment of \(\mathbf{RM}^t\). The main result states that \(\mathsf{PSM}\) is primitive, meaning that every subquasivariety is a variety, and therefore \(\mathsf{PSM}\) is structurally complete. This implies that the full negation-free fragment of \(\mathbf{RM}^t\) is hereditarily structurally complete. In contrast, the variety of Sugihara monoids, hence also \(\mathbf{RM}^t\), is structurally incomplete, revealing negation to be the sole cause of this.