Partial algebra

{{Short description|Algebraic structure}} In abstract algebra, a '''partial algebra''' is a generalization of universal algebra to partial operations.<ref name="Burmeister1993">{{cite book|editor1=Ivo G. Rosenberg |editor2=Gert Sabidussi|title=Algebras and Orders|year=1993|publisher=Springer Science & Business Media|isbn=978-0-7923-2143-9|author=Peter Burmeister|chapter=Partial algebras—an introductory survey | pages=1–70}}</ref><ref name=GAG>{{cite book|author=George A. Grätzer|author-link=George Grätzer|title=Universal Algebra|url=https://archive.org/details/isbn_9780387774862|url-access=registration|year=2008|publisher=Springer Science & Business Media|isbn=978-0-387-77487-9|at=Chapter 2. Partial algebras|edition=2nd}}</ref>

==Example(s)== * partial groupoid * field — the multiplicative inversion is the only proper partial operation<ref name="Burmeister1993"/> * effect algebras<ref>{{Cite journal | doi = 10.1007/BF02283036| title = Effect algebras and unsharp quantum logics| journal = Foundations of Physics| volume = 24| issue = 10| pages = 1331| year = 1994| last1 = Foulis | first1 = D. J.| last2 = Bennett | first2 = M. K.| bibcode = 1994FoPh...24.1331F| hdl = 10338.dmlcz/142815| s2cid = 123349992| hdl-access = free}}</ref>

==Structure== There is a "Meta Birkhoff Theorem" by Andreka, Nemeti and Sain (1982).<ref name="Burmeister1993"/> === Relational systems === Operations and partial operations may be written as finitary relations, where there is no requirement of totality. "A relational system <math>\mathfrak{A}</math> is a pair <''A'', ''R''>, where ''A'' is a non-void set and ''R'' is a family of (finitary) relations on ''A''."<ref name=GAG/>{{rp|8}}

Though relational systems have greater generality than algebras and partial algebras, they do not have the rich theory of the algebras.<ref>Richard S. Pierce (1968) ''Introduction to the Theory of Abstract Algebras'', page 17</ref> For example, defining a subalgebra of a relational system is not straight forward.<ref>Pierce page 28</ref>

== References == {{reflist}}

==Further reading== *{{cite book|author=Peter Burmeister|title=A Model Theoretic Oriented Approach to Partial Algebras|year=2002|orig-year=1986|citeseerx=10.1.1.92.6134}} * {{cite book|author=Horst Reichel|title=Structural induction on partial algebras|year=1984|publisher=Akademie-Verlag}} * {{cite book|author=Horst Reichel|title=Initial computability, algebraic specifications, and partial algebras|year=1987|publisher=Clarendon Press|isbn=978-0-19-853806-6}}

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Category:Algebraic structures