Abstract
We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA's is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the representation of MBA's as powerset algebras of certain directed graphs with a set of "marked" points. It is shown that there are only countably many varieties of MBA's, all are generated by their finite members, and all have finite equational axiomatisations classifying them into fourteen kinds of variety. The universal theory of each variety is decidable. Finitely generated MBA's are shown to be finite, with the free algebra on r generators having exactly, elements. An explicit procedure is given for constructing this freely generated algebra as the powerset algebra of a certain marked graph determined by the number r.
| Original language | English |
|---|---|
| Pages (from-to) | 1-40 |
| Number of pages | 40 |
| Journal | Studia Logica |
| Volume | 96 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2010 |
| Externally published | Yes |
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Keywords
- atom
- basic quantifier/algebra
- bounded existential quantifier
- bounded graph
- bounded morphism
- complex algebra
- finitely-generated
- free algebra
- free logic
- functional algebra
- model
- monadic algebra
- point-generated
- semantically complete logic
- simple algebra
- special algebra
- variety
- virtual ideal
ASJC Scopus subject areas
- Logic
Cite this
Monadic Bounded Algebras. / Akishev, Galym; Goldblatt, Robert.
In: Studia Logica, Vol. 96, No. 1, 2010, p. 1-40.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Monadic Bounded Algebras
AU - Akishev, Galym
AU - Goldblatt, Robert
PY - 2010
Y1 - 2010
N2 - We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA's is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the representation of MBA's as powerset algebras of certain directed graphs with a set of "marked" points. It is shown that there are only countably many varieties of MBA's, all are generated by their finite members, and all have finite equational axiomatisations classifying them into fourteen kinds of variety. The universal theory of each variety is decidable. Finitely generated MBA's are shown to be finite, with the free algebra on r generators having exactly, elements. An explicit procedure is given for constructing this freely generated algebra as the powerset algebra of a certain marked graph determined by the number r.
AB - We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA's is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the representation of MBA's as powerset algebras of certain directed graphs with a set of "marked" points. It is shown that there are only countably many varieties of MBA's, all are generated by their finite members, and all have finite equational axiomatisations classifying them into fourteen kinds of variety. The universal theory of each variety is decidable. Finitely generated MBA's are shown to be finite, with the free algebra on r generators having exactly, elements. An explicit procedure is given for constructing this freely generated algebra as the powerset algebra of a certain marked graph determined by the number r.
KW - atom
KW - basic quantifier/algebra
KW - bounded existential quantifier
KW - bounded graph
KW - bounded morphism
KW - complex algebra
KW - finitely-generated
KW - free algebra
KW - free logic
KW - functional algebra
KW - model
KW - monadic algebra
KW - point-generated
KW - semantically complete logic
KW - simple algebra
KW - special algebra
KW - variety
KW - virtual ideal
UR - http://www.scopus.com/inward/record.url?scp=77955184935&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77955184935&partnerID=8YFLogxK
U2 - 10.1007/s11225-010-9269-z
DO - 10.1007/s11225-010-9269-z
M3 - Article
VL - 96
SP - 1
EP - 40
JO - Studia Logica
JF - Studia Logica
SN - 0039-3215
IS - 1
ER -