# Questions tagged [monoid]

A monoid is an algebraic structure with a single associative binary operation and an identity element.

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### Commutative monoids with "bottom"

Is anyone aware of any existing terminology (and/or research) on the topic of the following structure, like where else it might appear, or what constraints can be placed on f for commutativity : ...
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### Is every associative $n$-ary operation with an identity element induced by a monoid?

Given any $n$-ary operation $*$ on a set $X$, an identity element for $*$ is an element $e \in X$ such that $x*e*e*...*e=e*x*e*e*...*e=...=e*e*...*e*x=x$ ($n-1$ $e$s in each product) for all $x \in X$....
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### Can you, given any semigroup, define an identity element to make it a monoid

I'm wondering if I can "make up" an identity element, like so: I can define an element I such that any element x + I is equal to x, i.e.: I can redefine my set as [the old set] union with {I}...
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### Notation for the product of all elements of a finite commutative subset in a monoid. [duplicate]

Let $M = (A,*)$ be a monoid, $fcs(M)$ be the set of all commutative subsets of $M$ (a subset $S$ of $M$ is commutative iff for all $a,b \in S$ holds $a*b=b*a$). Let $S \in fcs(M)$. Is there a ...
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Let $(L, \cdot)$ be a conmutative monoid with an identity $e$ and let $n \geq 1$ be a natural number. Show the following function is bijective. $F: Hom_{Monoid}(\mathbb{N}^n, L) \to L^n$ where $F(\... 9 votes 1 answer 247 views ### Prefixes of a word multiplying to the identity in a free group Let$A$be a finite alphabet, and let$w \in (A \cup A^{-1})^\ast$be a freely reduced word over the alphabet$A$and formal inverse symbols$A^{-1}$. Suppose$w$is non-empty. Can there ever be non-... 1 vote 0 answers 43 views ### Algorithm to check if a vector is in a finitely generated monoid Suppose we are given a sequence of integral vectors$\alpha_1,\alpha_2,\dots,\alpha_m\in\mathbb{Z}^n$. For some$\beta\in\mathbb{Z}^n$, is there an effective algorithm to check if $$\beta\in\mathbb{N}\... • 371 -1 votes 1 answer 73 views ### Infintie monoids satisfying a relation [closed] Let be A a set. It is endowed with one internal composition law which is also associative, let's say \cdot. There are two elements a_1, \: a_2\in A, such that:$$a_1x^na_2=x,\forall\: x\in A$$... • 965 1 vote 1 answer 58 views ### Left inverse in monoid, left dual in monoidal category, and uniqueness The notion of monoidal category is a categorification of the notion of monoid. If M is a monoid, consider the monoidal category Vec_M (of M-graded vector spaces over a field \mathbb{k}). If an ... • 5,427 3 votes 1 answer 55 views ### Inverse of ab in monoid implies a and b have inverses? Let a, b be elements in a monoid such that ab has an inverse. Is it true that a and b have inverses? Prove this if true or give a counterexample if false. I believe this is false because let ... 1 vote 2 answers 34 views ### Do sets of commuting elements having conjugates in a commutative submonoid have a single conjugating element? It is known that any set of commuting diagonalizable matrices is simultaneously diagonalizable. So, it would be nice to ask the following generalization: Given a monoid M (not necessarily a group), ... • 7,965 1 vote 0 answers 40 views ### Bitstring algebra, monoid with concatenation and bitwise boolean operations Given the free monoid on \{0,1\} that is the set of all finite, possibly empty binary strings.$$\langle \rangle, \langle 0\rangle, \langle 1\rangle, \langle 01\rangle \in \{0,1\}^*$$with the ... • 146 13 votes 2 answers 663 views ### Can all squares in a free group be made from squares in the free monoid? Here's a question I thought of, that I don't know the answer to. Let$F_2$be the free group on$\{a,b\}$, and$F_2^+$be the subset where all the exponents are positive. For a set$S$, let$S^2=\{g^2:...
Let $M$ be a commutative monoid and $m,x \in M$. Let $m^{-1}x$ be the set of elements $t$ such that $mt=x$ and suppose it not empty. Let finally $c$ be a cancellative element. Multiplication by $c$ ...