I usually explain monoids through glasses of beer, Domain modeling but here I will use plumbery pipes. For example, let's define a set of pipes this way: with monoids ”The set of pipes with a hole this size” Cyrille Martraire, Arolla, 2018 It is unlucky the word “Monoid” is so awkward, as it represents a concept that is both ubiquitous and actually simple. Given this definition of our set, now we can test if various elements belong to it or not. Monoids are all about *composability*. It's about having abstractions in the small that compose infinitely in the large. You are already familiar with many cases of typical monoidal composeablity, such as everything *group-by*, or everything *reduce*. What I want to emphasize is that Monoids happen to be frequent On top of this set, we define an operation, that we occurrences in business domains, and that you can call "combine", “append”, "merge", or "add", should spot them and exploit them. that takes two elements and combines1 them. Since I’ve first talked about my enthusiasm for monoids at conferences around the world, I’ve received multiple positive feedbacks of concrete situations where monoids helped teams redesign complicated parts of their domain models into something much simpler. And they would then say that the new design is “more elegant”, with a smile. One funny thing we notice now is that given two Just one important warning: please don’t confuse elements from the set, the result is always... in the monoids with monads. Monoids are much easier to set too! That sounds like nothing, but that's what understand than monads. For the rest of the text, Closure of Operations is all about: the set is closed we will keep monads out of scope, to focus solely under this operation. It matters. We have a little on monoids and their friends. system that's all about itself, always. That's cool. But there's more to it. If you first combine the first What are Monoids? two pipes together, then combine the result with the third pipe, or if you first combine the last two pipes Monoids come from a part of mathematics called then you combine the first pipe with it, then you end abstract algebra. It's totally intimidating, but it really up with the exact same result. We're not talking doesn't have to, at least not for monoids, which are about changing the ordering of the pipes, just the creatures so simple that kids pretty much master ordering of combining them. It's like putting the them by age of 2 (without knowing of course) parenthesis anywhere doesn't change the result. This is called Associativity, and is important. So what are monoids? A monoid is a mathematical structure. First we start with a simple set. A set is just a criterion to decide if elements belong or not to the set. Yes it’s a bit circular a definition but you get the idea. 1 In the world of pipes it could be named ‘weld”. ​ So what? So when I started explaining monoids to people I also explained it to my wife. She instantly got it, but then asked: what for? So I was the one wondering for a few seconds. Why does it matter to me? I believe it's all about dealing By the way changing the ordering of the pipes with encapsulating some diversity inside the would be called Commutativity, and we have it if structure. they all are identical, but this property is not as frequent. You probably know the old joke in programming: So we have a set of elements, an operation with results that are all in the set, and that is associative. There are only three numbers in To really be a monoid, you need one more thing: programming: 0, 1, and MANY. you have to define one more kind of pipe, that is invisible, so I can't show it to you. But believe me, it That's so true. But this also illustrates a very exists, because as a developer I can create the common kind of diversity we face all the time: the system the way I prefer it to be. This special singular, the plural, and the absence. Monoids element belongs to the set: it has a hole the right naturally represent the singular, with the elements size. So whenever I combine this special element of the set. It also deals with the plural, thanks to the combine operation that can take a plural and turn it with any other, the result is just the other element ​ itself. It doesn't change anything. That's why we call back into an element as usual. And the neutral it the neutral, or identity element. element takes care of the absence case, and it also belongs to the set. So monoids encapsulate this And voilà! A set, an operation, closure of this diversity inside their structure, so that from the operation, associativity, and neutral element: that's outside you don't have to care about it. That's a the formal definition of a monoid! great idea to fight the battle against complexity. Ok at this point, you are perhaps like: In the wild real life problems that we have, if we're not careful, we have to deal with various concepts, each potentially having their own diversity of being either singular, plural, or nothing. In the worst case, you'd end up with the cartesian product of all cases. It doesn't scale well. But stay with me, we'll see how this really closely relates to your daily job. Once you encapsulate this diversity inside a monoid for each concept, then you only have one case for All this is not difficult. Still, such a simple thing is a each, and together it remains one single case. key to very complex behaviors. It's also the key to infinite scalability of space (think Hadoop), and the key to infinite incremental scalability (think Storm). There's one joke in Big Data circles: If you're doing Big Data and you don't know what If you apply that often, then you can deal with high an abelian group is, then you do it wrong! levels of complexity with ease. Monoids help scale in complexity. In fact, if you're comfortable in It's all about *composability*, which is highly object-oriented programming, you may recognize desirable pretty much everywhere. something familiar you're doing already: for a given interface, I often find myself using the Composite pattern to deal with the plural, and the NullObject Implementing monoids in pattern to deal with the absence. These patterns help deal with singular, plural and absence your usual programming consistently, so that the caller code doesn't even have to know. That's similar in purpose. language So how do we implement monoids in plan Java code? Monoids are typical Functional Programming; In Functional Programming everything is a value; Therefore: Monoids are values! That's a solid proof that monoid are value objects. Seriously though they do have to be value objects, Examples Please! i.e. immutable and equality by value. But monoid objects don't have to be anemic, with just data. You already know a lot of monoids in your They are supposed to have behavior, and in programming language: particular behavior that compose, like lengths, where we want to be able to write: "18 m + 16 m = ● Integer with addition: Integers are closed ​ ​ ​ 34 m". The corresponding code for this method under addition: int + int = int. They're would be: associative: (3+5)+2=3+(5+2), and their neutral element is 0, since any integer plus public Length add(Length other){ zero gives the same integer. return new Length( ● Lists with list append operation: List + List ​ ​ ​ value + other.value); = List is closed under this appending, } which is associative: (a )+(b ,c)=(a , b)+(c). And the empty list is the neutral element This add() method returns a new instance, as value here. objects should do. It must not perform any ● A special case of lists, Strings with ​ ​ side-effect, as advocated by the DDD concatenation: "hello" + "world" is a string ​ "Side-Effect-Free Functions" pattern. Immutability too. It's associative: "cy"+"ri"+"lle", and the and Side-Effect-Free Functions together are good neutral element is the empty string. taste! That should be your default style of programming, unless you really have to do Note that integers can also form a monoid with the otherwise. multiplication operation, in which case the neutral element would be 1. But natural integers with In addition, being immutable and side-effect-free subtraction do not form a monoid, because 3 - 5 is means that testing is a no-brainer: just pass data in, not in the set of natural integers but it would in the set of integers. and assert the result out. Nothing else can happen to make it more complicated. Cashflows and sequences of Monoids in domain cashflows modeling Now that we have a money amount, we can make it into a cashflow, by adding a date: Domain-specific lists, like a mailing list defined as a list of emails addresses, can form a monoid at least (EUR, 25, TODAY) twice, once with the union operation, and a second + (EUR, 30, TODAY) time with the intersection operation. The neutral = (EUR, 55, TODAY) element would be nobody() in the former case, and everybody() in the latter case. Note the naming of And again we could throw an exception if the dates the neutral elements that is domain-specific, instead don't match.