Distributive laws in residuated binars

Abstract

In residuated binars there are six non-obvious distributivity identities of ·, /, \ over ∧, ∨. We show that in residuated binars with distributive lattice reducts there are some dependencies among these identities; specifically, there are six pairs of identities that imply another one of these identities, and we provide counterexamples to show that no other dependencies exist among these.Mathematics Subject Classification. 06F05, 03G10, 08B15.

Mining counterexamples for wide-signature algebras with an Isabelle server

“…Furthermore, by exhibiting some small countermodels (of size 4 and 5), [4] shows that the implications announced in the previous theorem completely characterize all interdependencies among (2)- (7) in the presence of lattice distributivity. [4] mentions the case without lattice distributivity as an open question.…”

“…This has found applications everywhere from establishing non-trivial categorical equivalences [5] to obtaining decidability results for models of program execution [13]. These identities are interdependent, and [4] establishes the following:…”

“…This contribution offers a case study in computer-assisted counterexample construction for algebras of wider signature. Our case study concerns residuated binars [4], each of which consists of an algebra A = (A, ∧, ∨, •, \, /) such that (A, ∧, ∨) is lattice, (A, •) is a set with a binary operation, and for all x, y, z ∈ A,…”

Priestley duality for MV-algebras and beyond

“…For a binary order type ε = (ε 1 , ε 2 ), a double quasioperator of type ε is a map f : A ε 1 × A ε 2 → A that preserves both meet and join in each coordinate (for more information, see [17] and [18], which provide a study of canonical extensions of double quasioperators of arbitrary, not-necessarily-binary order type. See also [8] for an algebraic study of equational conditions defining residuated double quasioperators. )…”

“…One approach, which offers at least a theoretical advantage, is extended Priestley duality for so-called double quasioperator algebras [17,18]. The latter comprise a huge class of lattice-ordered algebraic structures, including MV-algebras and, more generally, semilinear residuated binars (see, e.g., [8]). For these, first-order dual conditions are guaranteed under the condition that we double the non-lattice operations of arity two or higher.…”