diff --git a/.obsidian/community-plugins.json b/.obsidian/community-plugins.json
index dc1b1be..7549424 100644
--- a/.obsidian/community-plugins.json
+++ b/.obsidian/community-plugins.json
@@ -5,5 +5,6 @@
   "smart-second-brain",
   "local-gpt",
   "companion",
-  "pdf-plus"
+  "pdf-plus",
+  "obsidian-latex-suite"
 ]
\ No newline at end of file
diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index 8b81bd4..92f95ff 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -21,8 +21,23 @@
               "icon": "lucide-file",
               "title": "6a - Alternatives to Atomicity"
             }
+          },
+          {
+            "id": "149ad8d7dea2048d",
+            "type": "leaf",
+            "state": {
+              "type": "markdown",
+              "state": {
+                "file": "Concurrent Systems/notes/7- MUTEX-free concurrency.md",
+                "mode": "source",
+                "source": false
+              },
+              "icon": "lucide-file",
+              "title": "7- MUTEX-free concurrency"
+            }
           }
-        ]
+        ],
+        "currentTab": 1
       },
       {
         "id": "545279cecc435235",
@@ -37,9 +52,9 @@
               "state": {
                 "file": "Concurrent Systems/slides/class 6.pdf",
                 "page": 9,
-                "left": -23,
-                "top": 475,
-                "zoom": 0.6805225653206651
+                "left": -30,
+                "top": 442,
+                "zoom": 0.5201900237529692
               },
               "icon": "lucide-file-text",
               "title": "class 6"
@@ -102,8 +117,7 @@
       }
     ],
     "direction": "horizontal",
-    "width": 307.5,
-    "collapsed": true
+    "width": 307.5
   },
   "right": {
     "id": "bc4b945ded1926e3",
@@ -218,12 +232,13 @@
       "companion:Toggle completion": false
     }
   },
-  "active": "105f7a61963f10e7",
+  "active": "149ad8d7dea2048d",
   "lastOpenFiles": [
-    "Concurrent Systems/slides/class 6.pdf",
+    "Concurrent Systems/notes/Pasted image 20250324082534.png",
+    "Concurrent Systems/notes/7- MUTEX-free concurrency.md",
     "Concurrent Systems/notes/6a - Alternatives to Atomicity.md",
-    "Pasted image 20250324082545.png",
-    "Pasted image 20250324082534.png",
+    "Concurrent Systems/slides/class 6.pdf",
+    "Concurrent Systems/notes/Pasted image 20250324082545.png",
     "Concurrent Systems/notes/6 - Atomicity.md",
     "Concurrent Systems/notes/5 - Software Transactional Memory.md",
     "Concurrent Systems/notes/4b - Monitors.md",
@@ -264,7 +279,6 @@
     "Foundation of data science/notes/5 Neural Networks.md",
     "Foundation of data science/notes/8 Variational Autoencoders.md",
     "Foundation of data science/notes/3 Logistic Regression.md",
-    "Foundation of data science/notes/3.1 Multi Class Logistic Regression.md",
     "Senza nome.canvas"
   ]
 }
\ No newline at end of file
diff --git a/Concurrent Systems/notes/6a - Alternatives to Atomicity.md b/Concurrent Systems/notes/6a - Alternatives to Atomicity.md
index 6656723..2502a8e 100644
--- a/Concurrent Systems/notes/6a - Alternatives to Atomicity.md	
+++ b/Concurrent Systems/notes/6a - Alternatives to Atomicity.md	
@@ -32,3 +32,9 @@ Hence, we have two histories that are sequentially consistent but whose composit
 	- if the history is complete, we must have exactly one of these events for transaction
 - a sequential history is formed by committed transactions only
 
+**Def:** a complete history $\hat{H}$ is **serializable** if there exists a sequential history $\hat{S}$ s.t.
+1. $\forall X : \hat{S}|_X \in semantics(X)$
+2. $S = \{e \in H : e \in t \ in committedTrans(\hat{H})\}$
+3. $\to_{trans} \subseteq \to_{S}$ where $\to_{trans}$ is defined like $\to_{proc}$ in sequential consistency.
+
+It is a more general notion than linearizability, **but it is not compositional** (consider the previous example but with transactions instead of processes).
\ No newline at end of file
diff --git a/Concurrent Systems/notes/7- MUTEX-free concurrency.md b/Concurrent Systems/notes/7- MUTEX-free concurrency.md
new file mode 100644
index 0000000..e69de29
diff --git a/Pasted image 20250324082534.png b/Concurrent Systems/notes/Pasted image 20250324082534.png
similarity index 100%
rename from Pasted image 20250324082534.png
rename to Concurrent Systems/notes/Pasted image 20250324082534.png
diff --git a/Pasted image 20250324082545.png b/Concurrent Systems/notes/Pasted image 20250324082545.png
similarity index 100%
rename from Pasted image 20250324082545.png
rename to Concurrent Systems/notes/Pasted image 20250324082545.png