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Monadic NM-algebras [article]

Jun Tao Wang, Xiao Long Xin, Peng Fei He
2017 arXiv   pre-print
Also, we obtain some conditions under which monadic NM-algebras become monadic Boolean algebras.  ...  Finally, we present monadic NM-logic and prove the (chain) completeness of monadic NM-logic based on monadic NM-algebras.  ...  Monadic NM-algebras In this section, we introduce and investigate monadic NM-algebras. Then, we prove that monadic NM-algebras faithfully the axioms on quantifiers in monadic predicate NM logic.  ... 
arXiv:1709.04832v1 fatcat:5a7z5knijvelnhvubi6345btx4

Page 267 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 9, Issue 2 [page]

1958 American Mathematical Society. Proceedings of the American Mathematical Society  
The monadic algebra A above is free on the r generators {pi,- ++, Pe}. A has nm atoms, and so has Qum — a2"2?—; elements. We mention, without proof, a few of the properties of this algebra, A.  ...  1958] FINITE MONADIC ALGEBRAS 267 hence 3S, =NB(S,) Cie We now gather our results. By Theorem 1, Lemma 5, and Lemma 4, {p:1,---, pe} is a monadic generating set for A.  ... 

Page 7320 of Mathematical Reviews Vol. , Issue 2003j [page]

2003 Mathematical Reviews  
See also 20102, 54033, 68148. 06D Distributive lattices 2003j:06010 06D20 03G25 08B25 Bezhanishvili, Guram (1-NMS; Las Cruces, NM); Harding, John (1-NMS; Las Cruces, NM) Functional monadic Heyting algebras  ...  (English summary) Algebra Universalis 48 (2002), no. 1, 1-10. Monadic Heyting algebras were introduced as the algebraic coun- terparts of the one-variable fragment of intuitionistic logic.  ... 

Finite Monadic Algebras

Hyman Bass
1958 Proceedings of the American Mathematical Society  
The monadic algebra A above is free on the r generators {pi, ■ ■ • , pr}. A has nm atoms, and so has r C2r-1} 2»m _ 212 2 1 elements.  ...  A monadic algebra A is said to be free on the set of generators GCA if (i) G generates A, and (ii) any map /o of G into a monadic algebra B can be extended to a monadic homomorphism f:A->B.  ...  (3.2) We observe, before proceeding, that the algebra A has evidently nm atoms, (one element subsets of S).  ... 
doi:10.2307/2033149 fatcat:phhisz7xo5clhoj3ytp26lp7ym

Finite monadic algebras

Hyman Bass
1958 Proceedings of the American Mathematical Society  
The monadic algebra A above is free on the r generators {pi, ■ ■ • , pr}. A has nm atoms, and so has r C2r-1} 2»m _ 212 2 1 elements.  ...  A monadic algebra A is said to be free on the set of generators GCA if (i) G generates A, and (ii) any map /o of G into a monadic algebra B can be extended to a monadic homomorphism f:A->B.  ...  (3.2) We observe, before proceeding, that the algebra A has evidently nm atoms, (one element subsets of S).  ... 
doi:10.1090/s0002-9939-1958-0094308-5 fatcat:xqgw5gb2v5dirbju3ikhvihfkq

Solving Algebraic Equations Using Coalgebra

Federico De Marchi, Neil Ghani, Christoph Lüth
2003 RAIRO - Theoretical Informatics and Applications  
This paper presents a categorical approach to algebraic systems of equations which generalizes the traditional approach in two ways i) we define algebraic equations for locally finitely presentable categories  ...  Algebraic systems of equations define functions using recursion where parameter passing is permitted.  ...  Once this is understood, this would be a very exciting application. φ m (x 1 , . . . , x nm ) = t m (x 1 , . . . , x nm ) Lemma 3. 1 . 1 Let (T, η, µ) be a monad on C and F an endofunctor on C.  ... 
doi:10.1051/ita:2003021 fatcat:ksands7b4jfibpswdf2pomytg4

Functional distribution monads in functional-analytic contexts [article]

Rory B. B. Lucyshyn-Wright
2017 arXiv   pre-print
By employing the notion of commutant for enriched algebraic theories and monads, we define the functional distribution monad associated to a given functional-analytic context.  ...  The construction takes place within a categorical setting for generalized functional analysis, called a functional-analytic context, formulated in terms of a given monad or algebraic theory T enriched  ...  For all n, m ∈ N, let us denote by ι nm : T aff (n, m) → T (n, m) the equalizer of the morphisms φ nm = T (δ n , m) : T (n, m) → T (1, m) and ψ nm = Proposition 8.2.  ... 
arXiv:1701.08152v3 fatcat:tn3wd45hwveilgtwfkzaiqlwzi

Relating computational effects by ⊤⊤-lifting

Shin-ya Katsumata
2013 Information and Computation  
We consider the problem of establishing a relationship between two interpretations of base type terms of a λ c -calculus extended with algebraic operations.  ...  the global counter monad.  ...  It has the inverse Gef T : Alg(T , D, C) → C(C, T D) given by Gef T (α) = nm −1 C,T D (α D • nm D,T D (η D )). Let T ′ = (T ′ , η ′ , µ ′ , θ ′ ) be another strong monad on C.  ... 
doi:10.1016/j.ic.2012.10.014 fatcat:3dfxgbt5k5eo5o6ovrzvip7vae

Relating Computational Effects by ⊤ ⊤-Lifting [chapter]

Shin-ya Katsumata
2011 Lecture Notes in Computer Science  
We consider the problem of establishing a relationship between two interpretations of base type terms of a λ c -calculus extended with algebraic operations.  ...  the global counter monad.  ...  It has the inverse Gef T : Alg(T , D, C) → C(C, T D) given by Gef T (α) = nm −1 C,T D (α D • nm D,T D (η D )). Let T ′ = (T ′ , η ′ , µ ′ , θ ′ ) be another strong monad on C.  ... 
doi:10.1007/978-3-642-22012-8_13 fatcat:of4pmpczifdhjk6thzwgjwnj3u

Page 550 of Mathematical Reviews Vol. 51, Issue 2 [page]

1976 Mathematical Reviews  
In the case of the monadic category Vect, the Eilenberg-Moore algebra structure is explored in detail to show how it yields the usual algebra structure of a vector space.  ...  Linton’s (unpublished) resuit that Ban is equivalent to the Eilenberg-Moore category of algebras of the monad is given.  ... 

Lawvere theories, finitary monads and Cauchy-completion

Richard Garner
2014 Journal of Pure and Applied Algebra  
We also account for other phenomena from the enriched viewpoint: the equivalence of the algebras for a finitary monad with the models of the corresponding Lawvere theory; the functorial semantics in arbitrary  ...  categories with finite powers; and the existence of left adjoints to algebraic functors.  ...  The tensor of X (n) by y m is X (nm) , as witnessed by the element i = (π 1 , . . . , π nm ) ∈ LM m (X (n) , X (nm) ) = M nm (X, X) nm .  ... 
doi:10.1016/j.jpaa.2014.02.018 fatcat:fr5zkiw5rrg4rockmrwevdcm6a

Graded Algebraic Theories [article]

Satoshi Kura
2020 arXiv   pre-print
We provide graded extensions of algebraic theories and Lawvere theories that correspond to graded monads.  ...  We prove that graded algebraic theories, graded Lawvere theories, and finitary graded monads are equivalent via equivalence of categories, which extends the equivalence for monads.  ...  The inverse of the unit η T is the family of identity functions 1 : F T nm → F T nm.  ... 
arXiv:2002.06784v2 fatcat:qzvoodkv4vdl3cajfaws766qk4

Lawvere theories, finitary monads and Cauchy-completion [article]

Richard Garner
2013 arXiv   pre-print
We also account for other phenomena from the enriched viewpoint: the equivalence of the algebras for a finitary monad with the models of the corresponding Lawvere theory; the functorial semantics in arbitrary  ...  categories with finite powers; and the existence of left adjoints to algebraic functors.  ...  The tensor of X (n) by y m is X (nm) , as witnessed by the element i = (π 1 , . . . , π nm ) ∈ LM m (X (n) , X (nm) ) = M nm (X, X) nm .  ... 
arXiv:1307.2963v1 fatcat:peplpb2azbhmdcb7dxqkmavaa4

Graded Algebraic Theories [chapter]

Satoshi Kura
2020 Lecture Notes in Computer Science  
We provide graded extensions of algebraic theories and Lawvere theories that correspond to graded monads.  ...  We prove that graded algebraic theories, graded Lawvere theories, and finitary graded monads are equivalent via equivalence of categories, which extends the equivalence for monads.  ...  Monads correspond to algebraic theories [5] .  ... 
doi:10.1007/978-3-030-45231-5_21 fatcat:e26xprsfgvey5ktqn6xduav6em

Page 5415 of Mathematical Reviews Vol. , Issue 96i [page]

1996 Mathematical Reviews  
Loring (1-NM-S; Albuquerque, NM) 961:46084 46L85 19D55 22D25 46L05 Ji, Ronghui (1-INPI; Indianapolis, IN) Trace invariant and cyclic cohomology of twisted group C*-algebras. (English summary) J.  ...  In general uv :=[0] is not a monad and does not induce a vector space topology.  ... 
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