On universal algebra over nominal sets

We investigate universal algebra over the category Nom of nominal sets. Using the fact that Nom is a full reflective subcategory of a monadic category, we obtain an HSP-like theorem for algebras over nominal sets. We isolate a ‘uniform’ fragment of our equational logic, which corresponds to the nominal logics present in the literature. We give semantically invariant translations of theories for nominal algebra and NEL into ‘uniform’ theories, and systematically prove HSP theorems for models of these theories.

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Nominal (Universal) Algebra: Equational Logic with Names and Binding

Journal of Logic and Computation ◽

10.1093/logcom/exp033 ◽

2009 ◽

Vol 19 (6) ◽

pp. 1455-1508 ◽

Cited By ~ 38

Author(s):

M. J. Gabbay ◽

A. Mathijssen

Keyword(s):

Universal Algebra ◽

Equational Logic

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Fundamental relations and identities of fuzzy hyperalgebras

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-210994 ◽

2021 ◽

pp. 1-10

Author(s):

Narjes Firouzkouhi ◽

Abbas Amini ◽

Chun Cheng ◽

Mehdi Soleymani ◽

Bijan Davvaz

Keyword(s):

Universal Algebra ◽

Equivalence Relation ◽

Polynomial Function ◽

Equivalence Relations ◽

Fundamental Relation ◽

Term Function ◽

Biological Phenomena ◽

Definition Of

Inspired by fuzzy hyperalgebras and fuzzy polynomial function (term function), some homomorphism properties of fundamental relation on fuzzy hyperalgebras are conveyed. The obtained relations of fuzzy hyperalgebra are utilized for certain applications, i.e., biological phenomena and genetics along with some elucidatory examples presenting various aspects of fuzzy hyperalgebras. Then, by considering the definition of identities (weak and strong) as a class of fuzzy polynomial function, the smallest equivalence relation (fundamental relation) is obtained which is an important tool for fuzzy hyperalgebraic systems. Through the characterization of these equivalence relations of a fuzzy hyperalgebra, we assign the smallest equivalence relation α i 1 i 2 ∗ on a fuzzy hyperalgebra via identities where the factor hyperalgebra is a universal algebra. We extend and improve the identities on fuzzy hyperalgebras and characterize the smallest equivalence relation α J ∗ on the set of strong identities.

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Algebras Describing Pseudocomplemented, Relatively Pseudocomplemented and Sectionally Pseudocomplemented Posets

Symmetry ◽

10.3390/sym13050753 ◽

2021 ◽

Vol 13 (5) ◽

pp. 753

Author(s):

Ivan Chajda ◽

Helmut Länger

Keyword(s):

Universal Algebra ◽

Relational Structures ◽

Pseudocomplemented Poset ◽

The Given ◽

The Relationship ◽

Congruence Properties

In order to be able to use methods of universal algebra for investigating posets, we assigned to every pseudocomplemented poset, to every relatively pseudocomplemented poset and to every sectionally pseudocomplemented poset, a certain algebra (based on a commutative directoid or on a λ-lattice) which satisfies certain identities and implications. We show that the assigned algebras fully characterize the given corresponding posets. A certain kind of symmetry can be seen in the relationship between the classes of mentioned posets and the classes of directoids and λ-lattices representing these relational structures. As we show in the paper, this relationship is fully symmetric. Our results show that the assigned algebras satisfy strong congruence properties which can be transferred back to the posets. We also mention applications of such posets in certain non-classical logics.

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Categorification of algebraic quantum field theories

Letters in Mathematical Physics ◽

10.1007/s11005-021-01371-8 ◽

2021 ◽

Vol 111 (2) ◽

Author(s):

Marco Benini ◽

Marco Perin ◽

Alexander Schenkel ◽

Lukas Woike

Keyword(s):

Universal Algebra ◽

Finite Groups ◽

Physical Interpretation ◽

Field Theories ◽

Quantum Field Theories ◽

Quantum Field ◽

Algebraic Quantum

AbstractThis paper develops a concept of 2-categorical algebraic quantum field theories (2AQFTs) that assign locally presentable linear categories to spacetimes. It is proven that ordinary AQFTs embed as a coreflective full 2-subcategory into the 2-category of 2AQFTs. Examples of 2AQFTs that do not come from ordinary AQFTs via this embedding are constructed by a local gauging construction for finite groups, which admits a physical interpretation in terms of orbifold theories. A categorification of Fredenhagen’s universal algebra is developed and also computed for simple examples of 2AQFTs.

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