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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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Assume $\Gamma$ is an infinite set composed of formulas (of finite length), and $A$ is a formula (of finite length). For example, $\Gamma=\{x,y\sqcup z,x_1,x_2,x_3,\cdots\}$, $A=(x\sqcap y)\sqcup(x\...
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The background for my question is somewhat related to this; there a very interesting paper is provided, but the setting and examples are somewhat different. I can add any necessary background or ...
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Let $(R, +, 0, \cdot, 1)$ be a (unital) semiring equipped with a unary operation $f$ and a binary operation $\land$ such that $f(0) = 1$ $f(x + y) = f(x) f(y)$ $x \land y = y \land x$ $x \land (y \...
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The same post's presented on mathstackexchange, but I think this one is a research level question. I was preparing for my postgraduate exams and one of the questions concerned constructing non-...
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Let $V := \{v_0, v_1, \ldots\}$ be a countable set of variables and let $A := \mathbb{Z}[x, y]^{\oplus V}$ with $V \subseteq A$ its basis as a $\mathbb{Z}[x, y]$-module. Define $$m(a, b) := xa + yb$$ ...
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Given a variety (in the sense of universal algebra) $\mathscr{V}$ axiomatized by a finite set of equations $E$, say that $\mathscr{V}$ is consistently gappy iff it is consistent with $\mathsf{ZF}$ ...
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From my understanding a single sort term algebra $T_\Sigma(V)$ is a set of 'syntactic' objects generated over a signature $\Sigma$ of function names (with provided arities) and a set of variables $V$. ...
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The set of finite combinatorial games is the smallest set $\mathscr{G}$ such that $\{\vert\}\in\mathscr{G}$ and, whenever $A,B\subseteq\mathscr{G}$ are finite, we have $\{A\vert B\}\in\mathscr{G}$. ...
Noah Schweber's user avatar
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Let $S = \{z, i, p, d, u\}$ be a 5-element set with distinct elements. We seek to define a binary operation $\cdot : S \times S \to S$ satisfying: $\cdot$ is commutative. $i \in S$ is a two-sided ...
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Let $T$ and $T'$ be two equational theories, such that there is a isomorphism between the clones $\mathrm{Cl}(T)$ and $\mathrm{Cl}(T')$ associated with these theories. Are these theories $T$ and $T'$ ...
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This question is based on trying to provide an answer to another question of mine: Are there categories of incidence structures and projective geometries? The basic question is: If $T$ is a universal ...
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Given a small set of morphisms $M$ in a locally presentable category $C$, there exists an factorization system $(M^{rl}, M^r)$ generated by $M$. In the general case, the factorization is constructed ...
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Let $E$ be an equational theory (in the sense of universal algebra) in the language of lattices. Given a cardinal $\kappa$, say that $E$ is $\kappa$-cheap iff there is a set-sized complete lattice $L$ ...
Noah Schweber's user avatar
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A lot of famous compacta have interesting continuous algebraic structures. For example, $S^1$ can be viewed as the circle group. $2^\omega$ can be viewed as the ring of $p$-adic integers for any $p$. ...
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Preface: I have a bad habit of writing questions that are too general. This is a very specific question I encountered while trying to answer my own question here. Hopefully this focus will be ...
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This question is belatedly inspired by an answer of Keith Kearnes. Say that a relation algebra $\mathcal{A}=(A;0,1,\check{\Box}, \overline{\Box},I, \circ, \wedge,\vee)$ is unary-representable if ...
Noah Schweber's user avatar
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What is the profinite completion of the initial pointed magma? Okay, well, I should clarify what I mean by "What is". Clearly such a structure will be quite hard to describe nicely because ...
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In the context of finite groups and vector spaces, cancellation in direct sum (or direct product) isomorphisms is well-understood: i.e for finite dimensional vector space if $V \oplus W \cong V \oplus ...
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Every finitary algebraic category* is well-powered, simply because the monomorphisms are precisely the injective homomorphisms. Unfortunately, epimorphisms in the sense of category theory are not so ...
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Recall that any nonempty set $S$ with an associative binary operation is a semigroup. An involution semigroup is a pair $(S,{}^\star)$, where $S$ is a semigroup endowed with a unary operation $^\star$ ...
E W H Lee's user avatar
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Given two terms over the signature $(+,\times,\mathrm{factorial}(\cdot),0,1)$ with arbitrarily many variables, is there an algorithm to decide whether the two terms define the same function on $\...
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I asked this question on math stack exchange, but it didn't get any responses. So, I am asking it here. Suppose we are working in the signature of a single binary operation $*$. We are given a finite ...
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Let $\mathsf{T}$ be a finitary monad over the category of sets. Q. Under what conditions (possibly on the monad) the category of algebras $\mathsf{Alg(T)}$ is going to be balanced? I cannot find any ...
Ivan Di Liberti's user avatar
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$\DeclareMathOperator\Th{Th}$Two first order theories $T_1, T_2$ without relation symbols are said to be term-equivalent if there is a synonymy between them that only uses definitions of function ...
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I'm trying to understand congruence-permutability from the perspective of category theory. The most straightforward way to do this is to directly translate the condition that it is named for: $\...
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Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that $(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids $x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
Keith J. Bauer's user avatar
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By a well-known result of Bol (1937) and Bruck (1946), for any loop $X$ the following two identities are equivalent: B: $\forall x,y,z\in X\;\;x(y(xz))=((xy)x)z$ M: $\forall x,y,z\in X\;\;(xy)(zx)=(x(...
Taras Banakh's user avatar
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Let $f : S \to T$ be a morphism of algebraic theories. Such a morphism induces a monadic functor $f^* : \mathrm{Mod}(T) \to \mathrm{Mod}(S)$ (hence $f^*$ has a left adjoint). We may view $f$ ...
varkor's user avatar
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For a single-sorted algebraic theory $\mathcal{T}$ denote by $t_n$ the number of $\mathcal{T}$-algebras with $n$ elements (up to isomorphism). Is there an example for $\mathcal{T}$ such that ...
Martin Brandenburg's user avatar
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We have the idea of a prime ideal in a commutative ring $R$ but in universal algebra, we generalize the notion of ideal to that of a congruence. I’ve thought over the question of what a prime ...
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In his dissertation on "Functorial semantics of algebraic theories", Lawvere says in his introduction that "from the category (or more precisely from an underlying-set functor) we can ...
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$\newcommand\Eq{\mathrm{Eq}}$I asked this question on math stack exchange, here, but there were no comments or answers. So, I am asking it here on mathoverflow. Consider the signature of a single ...
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A variety of groups is a class of groups satisfying a specified set of equations. Equivalently, it is a class of groups that is closed under homomorphic images, subgroups, and direct products. A ...
Martin Brandenburg's user avatar
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Motivation There is a theory of smooth spaces (for example, diffeological spaces) as certain sheaves on the site $\mathrm{Cart}$, which is the "category of cartesian spaces" whose objects ...
andres's user avatar
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The title has it all: Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups? If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
Salvo Tringali's user avatar
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Consider the fragment of first-order logic with equality and universal quantification as the only logical symbols; we call this logic the logic of universal algebra. I am interested in languages $\...
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By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
Salvo Tringali's user avatar
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I asked this question on Math Stack Exchange a while ago, but no one has responded yet. So, I am asking it here. Consider the signature of a single binary operation $+$, and consider the set $Eq$ of ...
user107952's user avatar
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To motivate the question, consider the theory of rings. Define $x \parallel y$ to mean $\exists w \exists z .((x - y) z = w (x - y) = 1)$, or in words, "$x - y$ is a unit". Then $\parallel$ ...
Zhen Lin's user avatar
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I think I'm in the process of understanding something very subtle here, and I could use an expert's double check. So basically, my question is whether what I write is correct. (Non-finitary) GATs, ...
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Welcome to my first MathOverflow posting! This is a question about rings, all of them assumed to be both unital and associative. Let $f\colon R\to S$ be a map between rings such that $f(xy)=f(x)f(y)$ ...
Fred Wehrung's user avatar
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Given an algebra $\mathbf{A}$, a pair of congruences $ \alpha ,\beta \in Con(\mathbf {A})$ are said to permute when $ \alpha \circ \beta =\beta \circ \alpha$, and an algebra $\mathbf{A}$ is called ...
Arena's user avatar
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Let $\mathbf{Grp}$ be the category of groups. Given a subcategory $\mathscr{G}$ of $\mathbf{Grp}$ and $G\in\mathit{Ob}(\mathscr{G})$, a $\mathscr{G}$-extendible map on $G$ will here mean an assignment ...
Noah Schweber's user avatar
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Suppose $\mathscr{V}$ is a pseudovariety in a countable language $\Sigma$. Say that a minimal description of $\mathscr{V}$ (in the sense of Eilenberg/Schutzenberger rather than Reiterman) is a pair $(...
Noah Schweber's user avatar
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See Eilenberg/Schutzenberger, On pseudovarieties for background on pseudovarieties. I've phrased things in terms of pairs-of-sets to avoid some annoying language about multisets. Also, I'm aware that ...
Noah Schweber's user avatar
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Let $B$, $C$, and $D$ be abelian groups such that $\mathbb Z\times B$ is isomorphic to $C\times D$. Is there a group $E$ such that $C$ or $D$ is isomorphic to $\mathbb Z\times E$? Reference: page 263 ...
Tri's user avatar
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This is motivationally related to an earlier question of mine. Given a first-order theory $T$, let $\widehat{D}(T)$ be the semiring defined as follows: Elements of $\widehat{D}(T)$ are equivalence ...
Noah Schweber's user avatar
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Have the semigroups with the following cancellation property been studied? Property: Let $S$ be a semigroup and $x,y\in S$ such that $xz=yz,$ for all $z\in S,$ then $x=y$.
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Given a clone $\mathcal{C}$ over $\{\top,\perp\}$, let $\mathsf{FOL}^\mathcal{C}$ be the version of first-order logic with connectives from $\mathcal{C}$ in place of the usual Booleans. Given a clone $...
Noah Schweber's user avatar
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Given a clone $\mathcal{C}$ over the set $\{\top,\perp\}$, let $\mathsf{FOL}^\mathcal{C}$ be the version of first-order logic with (symbols corresponding to) elements of $\mathcal{C}$ replacing the ...
Noah Schweber's user avatar
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