# 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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### In which algebraic theories do 'free' and 'projective' coincide?

Free models of algebraic theories are always projective objects in the category of models, but the converse is not always true. For instance, some (actually, all) projective modules are direct ...
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### Is there a finite list of identites in the language of $(\mathbb{N},0,1,+,\times,\mathrm{gcd},\mathrm{lcm})$ that generates all the others?

Let $\Phi$ denote the set of all identities satisfied by $(\mathbb{N},0,1,+,\times,\mathrm{gcd},\mathrm{lcm}).$ Question. Is $\Phi$ finitely axiomatizable? If so, I'd like to see a list of ...
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### Most general form of Cayley's theorem?

For many classes of algebraic structures, there exists a family of structures such that any member of the class can be embedded in some member of the family (groups and symmetric groups, unital rings ...
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### Is there a systematic way of “discovering” an algebra from observations of its universe?

I am faced with the following situation: I have a finite set of some $m$ positive integers $Q^m \in \mathbb{N}$ These integers go through a series of $N$ possible black boxes that transform them. ...
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### A subalgebra of $P(M, \Delta)$ is a Peano Algebra

Let $M$ be any set and let $\Delta=(n_i)_{i \in I}$ be an algebraic type. Let $P=P(M, \Delta)$ be the algebra such that the generalized Peano Axioms hold: (P1) $f_{i}(a_0,\dots,a_{n-1}) \notin M$ (...
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### What can we say about generating subsets if all free subsets are finite

Definitions An (universal) algebra is a pair $\mathcal A=(A, (f_1,\dots, f_n))$ where $A$ is a non-empty set and $(f_1, \dots, f_n)$ is a family of finitary operations on $A$. The notation $o(f_i)$ ...
I read somewhere a discription of the complex Clifford algebra as a $C^*$-algebra, but I don't know where... Is the complex Clifford algebra the universal $C^*$-algebra generated by elements $1$ and $... 0answers 50 views ### Slick Definition of the Category of Cartesian Closed Categories I can produce elementary definitions by just inspecting the definition on nlab, but is there a readily available abstract definition? I vaguely remember seeing that they could be defined as algebras ... 0answers 28 views ### Non-roots of unity auxillary constants in a group? Let$A$be a set, together with a set$F$of n-ary operations on A, which may include constants of$A$as 0-ary operations. A set$G$of operations on$A$is said to be auxillary with respect to the ... 0answers 23 views ### what can we say if we just know the global section has a given universal algebra structure? Given an universal algebra A, how to reconstruct a topological space X with some universal (and natural) property (e.g initial,terminal...)in the category of topological spaces whose global sections ... 0answers 32 views ### Does$\mathbf{Tos}$generate$\mathbf{DLat}$as a variety? Let$\mathbf{DLat}$denote the variety of distributive lattices and let$\mathbf{Tos}$denote the subclass of$\mathbf{DLat}$consisting of the totally-ordered sets. Question. Does$\mathbf{Tos}$... 0answers 42 views ### Optimal object in category? Metric, objective function on category objects. Optimization over category? Is there notion about optimal object in category (that can be found by some algorithms, or - more importantly - that can be constructed (if unknown) by some algorithms), about metric and objective ... 0answers 87 views ### Existential and Universal Equivalence Proof Taking ¬∀x:X.r≡∃x:X.¬r Is there a way of actually formally proving this? Not implementing it but proving how to go from a negated universal quantifier to a an existential with a negated element... ... 0answers 37 views ### Are there any preservation theorems for quotients of subalgebras? Let$X$denote an algebraic structure. Then: Every subalgebra of$X$satisfies each quasi-identity that is satisfied by$X$. In other words, taking subalgebras preserves quasi-identities. Every ... 0answers 30 views ### What is the standard term for the property of an order-embedding$h \colon P \to P$such that for all$a \in P$,$a \leq h(a)$? In an essay, I want to talk about order-embeddings$h \colon P \to P$on a partially ordered set P such that for all$a \in P$,$a \leq h(a)$. Does somebody know the standard term for a function ... 0answers 49 views ### Question on free Boolean algebras Every Boolean algebra$A$is isomorphic to a field of set. In particular, if$A$is finite, then$A$is isomorphic to the power set of its atoms. Now, suppose that$A$is free Boolean algebra with 2 ... 0answers 28 views ### Example of relatively free lattice "A relatively free lattice with$n$generators has exactly$n$maximal sublattices, each obtained by removing one of the generators (which are doubly irreducible). Thus there exist arbitrarily large ... 0answers 33 views ### Question about “immediate” observation about finitely presentable objects The following is an excerpt from volume II of Borceux: Here$F$is the left adjoint to the forgetful functor$U:\mathsf{Mod}_\mathcal{T}\longrightarrow \mathsf{Set}$. I'm confused and can't manage ... 0answers 64 views ### Generators of free Boolean algebras Suppose$\mathfrak{A}$is a free Boolean algebra and$G$a countable set of free generators of$\mathfrak{A}$. What is the cardinality of$\mathfrak{A}$if$G$is countably infinite, but we only ... 0answers 59 views ### Universal Algebra: partial algebras and homomorphisms I'm looking for an accessible introductory text on universal algebra, one which discusses homomorphisms in the context of partial algebras. I have been recommended the Grätzer book, "Universal ... 0answers 164 views ### Definition of$\Omega$-algebra I'm studying universal algebra. I have this definiton: given a signature$\Omega$, an$\Omega$-algebra is comprised of a "carrier" for the algebra and an "interpretation" for every operation symbol. ... 0answers 130 views ### Is this a general structure for constructs? Here a construct is a category where the objects are sets and the morphisms are structure preserving functions. Common examples are groups, graphs and topological spaces. As far as I can see there is ... 0answers 46 views ### Is there accepted terminology for algebraic structures whose every subalgebra is free? Is there accepted terminology for algebraic structures whose every subalgebra is free? Examples: Any free group Any vector space More generally, any free module over a PID. In fact, this ... 0answers 83 views ### Congruence lattice of$N_5$I calculated the the congruence lattice of$N_5$using hit and trial and then verified it with Universal Algebra calculator. But I need to prove that it is the congruence lattice of$N_5$How should I ... 0answers 36 views ### Question on HSP and SHPS inquality. In the screenshots attached above George Bergman outlines his way of proving$HSP \ne SHPS$I understand the first definition as the group of affine transformations and each element of the group can ... 0answers 166 views ### Collections of Homomorphic (defined) structures via$f$Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism An Example similar to the construction I found was this: Lets take define ... 0answers 57 views ### What is the$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$. So, in trying to ... 0answers 81 views ### Congruence lattice of a partial algebra is algebraic A partial operation on a nonempty set$A$is a map$f:\mathrm{dom}(f,A)\to A$where$\mathrm{dom}(f,A)\subseteq A^n$for some$n\in\mathbb{N}$. A partial algebra is an ordered pair$(A,P)$where$A$... 0answers 74 views ### Which “conditions” generate subalgebras? While looking at this question I suddenly wondered about a more general question. Think of your favorite class of algebra (groups or rings, say), and forgive me for vocabulary mistakes while I try to ... 0answers 82 views ### Another way of saying that algebraic objects are isomorphic From a universal algebraic perspective, let's say we have two isomorphic groups. Then can I speak of their isomorphic nature by saying the binary operations of multiplication of the two groups are "... 0answers 89 views ### direct product of different algebras? Is it possible to define a "direct product" of two algebras with different signatures? For example, boolean lattice$\otimes$monoid? Perhaps we need to take some quotient to make sense of it (e.g Q ... 0answers 222 views ### amalgam of structures Trying to refine my question here. This is a response to the questions here: Homomorphisms between structures My objective is to take a set of$S-$structures and form an amalgam object out of that ... 0answers 37 views ### Does this algebra whose signature is (1,1) have a name? Let there be an algebra$(S,f,t)$with the laws: $$f(t(x)) = t(x) \\ t(f(x)) = t(x)$$ or, put another way, $$f \circ t = t \\ t \circ f = t.$$ Does that particular algebra have a name? Does a ... 0answers 31 views ### It has at most one absorbing element Let$G = (M, \circ )$be a groupoid and let$2^G = (2^M, \circ_K)$a groupoid ($\circ_K$is the Product of group subsets). How can show that$G$has at most one absorbing element? 0answers 69 views ### Equivalence of two definitions of complete distributivity I would like to know if the following alternative definitions of complete distributivity are equivalent. Let's begin with defining choice functions: For any set$S$and$U\in \mathcal{P}(\mathcal{P}(...
Suppose that you have a signature $\mathfrak{F}$ containing at least one constant symbol $f$ of arity $0$. How does the absolutely free algebra (genrated by $X$) interpret this symbol? I know that the ...