# Tagged Questions

The study of algebraic structures and properties applying to large classes of such structures. For example, ideas from group theory and ring theory are extended and considered for structures with other signatures (systems of basic or fundamental operations).

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### What is the $K$-free algebra for the class of implication algebras, over a finite set

I suppose the title is pretty self explanatory. I have been struggling with the concepts of $K$-free algebras, where $K$ is some class of same-type algebras, over some set $X$. So, in trying to ...
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If I have structure $(S, \cdot)$, where $\cdot$ has type $(2)$, i.e., $\cdot : S \times S \rightarrow S$ and $(S', \circ)$, where $\circ$ has type $(3)$, i.e., $\cdot : S' \times S' \times S' \... 1answer 62 views ### Existence of arbitrarily large ordinal subgroups in a group structure on a regular cardinal [duplicate] Suppose$\kappa$is an uncountable regular cardinal, and$(\kappa, \cdot, ^{-1}, e$) is a group. Prove that that$C = \{\alpha < \kappa: \alpha\, \textrm{is a subgroup of}\, \kappa)$is unbounded ... 1answer 129 views ### What is a simple axiomatisation of groups using division? I recall from an old exercise I did as an undergrad that groups can be axiomatised using division rather than multiplication: A group is a non-empty set equipped with a binary division operator / ... 4answers 421 views ### Preserving structures Category theory abstracts the notion of the preservation of structure by means of morphisms. Is there a description of what it means to preserve structure of different types of mathematical structures ... 4answers 274 views ### Any commutative associative operation can be extended to a function on nonempty finite sets This is a fact we use very frequently in general mathematics when we write such notations as$1+2+3+4$: since we know that$+$is commutative and associative, we can just "drop the parentheses" and ... 2answers 373 views ### In a slice category C/A of a category C over a given object A, What is the role of the identity morphism of A in C with respect to C/A In a slice category$C/A$of a category$C$over a given object$A$, what is the role of the$C$identity morphism,$A\to A$($1_A$), in$C/A$, particularly with respect to composition? I ... 1answer 330 views ### Property: closure under an binary operation We have the following definition: Definition: let$*_A:A^2 \rightarrow A$and$B \subseteq A $, with$B \neq \emptyset$,$B$is closed under$*_A$if$a *_A|_Bb \in B\forall a,b \in B$. Property: ... 1answer 122 views ### Algebraic substructure and restriction of a function I am reading the follow pdf: http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf in particular at pg 28 of pdf, and I think that, let$(A;f)$an algebric structure and$B \...
Sorry if this is a silly question. Define that a right-cancellative semigroup is a set $G$ together with an associative operation $*$ such that for all $a,b,x \in G$ it holds that \$ax=bx \Rightarrow ...