TY - JOUR
T1 - Residuation algebras with functional duals
AU - Fussner, Wesley
AU - Palmigiano, Alessandra
PY - 2019/12
Y1 - 2019/12
N2 - We employ the theory of canonical extensions to study residuation algebras whose associated relational structures are functional, i.e., for which the ternary relations associated to the expanded operations admit an interpretation as (possibly partial) functions. Providing a partial answer to a question of Gehrke, we demonstrate that functionality is not definable in the language of residuation algebras (or even residuated lattices), in the sense that no equational or quasi-equational condition in the language of residuation algebras is equivalent to the functionality of the associated relational structures. Finally, we show that the class of Boolean residuation algebras such that the atom structures of their canonical extensions are functional generates the variety of Boolean residuation algebras.
AB - We employ the theory of canonical extensions to study residuation algebras whose associated relational structures are functional, i.e., for which the ternary relations associated to the expanded operations admit an interpretation as (possibly partial) functions. Providing a partial answer to a question of Gehrke, we demonstrate that functionality is not definable in the language of residuation algebras (or even residuated lattices), in the sense that no equational or quasi-equational condition in the language of residuation algebras is equivalent to the functionality of the associated relational structures. Finally, we show that the class of Boolean residuation algebras such that the atom structures of their canonical extensions are functional generates the variety of Boolean residuation algebras.
KW - Canonical extensions
KW - Definability of functionality
KW - Residuation algebras
UR - http://www.scopus.com/inward/record.url?scp=85073225172&partnerID=8YFLogxK
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U2 - 10.1007/s00012-019-0613-5
DO - 10.1007/s00012-019-0613-5
M3 - Article
AN - SCOPUS:85073225172
SN - 0002-5240
VL - 80
SP - 1
EP - 10
JO - Algebra Universalis
JF - Algebra Universalis
IS - 4
M1 - 40
ER -