Reiterman’s Theorem on Finite Algebras for a Monad

Jiří Adámek; Liang-Ting Chen; Stefan Milius; Henning Urbat

doi:10.1145/3464691

Reiterman’s Theorem on Finite Algebras for a Monad

ACM Transactions on Computational Logic ◽

10.1145/3464691 ◽

2021 ◽

Vol 22 (4) ◽

pp. 1-48

Author(s):

Jiří Adámek ◽

Liang-Ting Chen ◽

Stefan Milius ◽

Henning Urbat

Keyword(s):

Formal Languages ◽

Technical Tool ◽

Algebraic Classification ◽

Finite Algebras ◽

Finite Products ◽

Indispensable Tool

Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’s theorem states that they precisely specify pseudovarieties, i.e., classes of finite algebras closed under finite products, subalgebras and quotients. In this article, Reiterman’s theorem is generalized to finite Eilenberg-Moore algebras for a monad T on a category D: we prove that a class of finite T -algebras is a pseudovariety iff it is presentable by profinite equations. As a key technical tool, we introduce the concept of a profinite monad T ^ associated to the monad T , which gives a categorical view of the construction of the space of profinite terms.

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Algebraic Classification of Field Equations

The General Theory of Relativity ◽

10.1007/978-1-4614-3658-4_7 ◽

2012 ◽

pp. 465-536

Author(s):

Anadijiban Das ◽

Andrew DeBenedictis

Keyword(s):

Field Equations ◽

Algebraic Classification

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Classification of representation-finite algebras and their modules

Handbook of Tilting Theory ◽

10.1017/cbo9780511735134.003 ◽

2010 ◽

pp. 15-30

Author(s):

Thomas Brüstle

Keyword(s):

Finite Algebras

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The Algebraic and Geometric Classification of Four Dimensional Superalgebras

10.26686/wgtn.16915645.v1 ◽

2021 ◽

Author(s):

◽

Aaron Armour

Keyword(s):

Classification Problem ◽

Associative Algebras ◽

New Variety ◽

Graded Algebras ◽

Finite Representation ◽

Algebraic Classification ◽

Irreducible Components ◽

Module Algebras ◽

Sweedler’S Hopf Algebra

The algebraic and geometric classification of k-algbras, of dimension fouror less, was started by Gabriel in “Finite representation type is open” [12]. Several years later Mazzola continued in this direction with his paper “Thealgebraic and geometric classification of associative algebras of dimensionfive” [21]. The problem we attempt in this thesis, is to extend the resultsof Gabriel to the setting of super (or Z2-graded) algebras — our main effortsbeing devoted to the case of superalgebras of dimension four. Wegive an algebraic classification for superalgebras of dimension four withnon-trivial Z2-grading. By combining these results with Gabriel’s we obtaina complete algebraic classification of four dimensional superalgebras. This completes the classification of four dimensional Yetter-Drinfeld modulealgebras over Sweedler’s Hopf algebra H4 given by Chen and Zhangin “Four dimensional Yetter-Drinfeld module algebras over H4” [9]. Thegeometric classification problem leads us to define a new variety, Salgn —the variety of n-dimensional superalgebras—and study some of its properties. The geometry of Salgn is influenced by the geometry of the varietyAlgn yet it is also more complicated, an important difference being thatSalgn is disconnected. While we make significant progress on the geometricclassification of four dimensional superalgebras, it is not complete. Wediscover twenty irreducible components of Salg4 — however there couldbe up to two further irreducible components.

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